1.1: The Science of Biology
Skills to Develop
- Identify the shared characteristics of the natural sciences
- Summarize the steps of the scientific method
- Compare inductive reasoning with deductive reasoning
- Describe the goals of basic science and applied science
What is biology? In simple terms, biology is the study of living organisms and their interactions with one another and their environments. This is a very broad definition because the scope of biology is vast. Biologists may study anything from the microscopic or submicroscopic view of a cell to ecosystems and the whole living planet (Figure \(\PageIndex{1}\)). Listening to the daily news, you will quickly realize how many aspects of biology are discussed every day. For example, recent news topics include Escherichia coli (Figure \(\PageIndex{2}\)) outbreaks in spinach and Salmonella contamination in peanut butter. Other subjects include efforts toward finding a cure for AIDS, Alzheimer’s disease, and cancer. On a global scale, many researchers are committed to finding ways to protect the planet, solve environmental issues, and reduce the effects of climate change. All of these diverse endeavors are related to different facets of the discipline of biology.
The Process of Science
Biology is a science, but what exactly is science? What does the study of biology share with other scientific disciplines? Science (from the Latin scientia , meaning “knowledge”) can be defined as knowledge that covers general truths or the operation of general laws, especially when acquired and tested by the scientific method. It becomes clear from this definition that the application of the scientific method plays a major role in science. The scientific method is a method of research with defined steps that include experiments and careful observation.
The steps of the scientific method will be examined in detail later, but one of the most important aspects of this method is the testing of hypotheses by means of repeatable experiments. A hypothesis is a suggested explanation for an event, which can be tested. Although using the scientific method is inherent to science, it is inadequate in determining what science is. This is because it is relatively easy to apply the scientific method to disciplines such as physics and chemistry, but when it comes to disciplines like archaeology, psychology, and geology, the scientific method becomes less applicable as it becomes more difficult to repeat experiments.
These areas of study are still sciences, however. Consider archeology—even though one cannot perform repeatable experiments, hypotheses may still be supported. For instance, an archeologist can hypothesize that an ancient culture existed based on finding a piece of pottery. Further hypotheses could be made about various characteristics of this culture, and these hypotheses may be found to be correct or false through continued support or contradictions from other findings. A hypothesis may become a verified theory. A theory is a tested and confirmed explanation for observations or phenomena. Science may be better defined as fields of study that attempt to comprehend the nature of the universe.
Natural Sciences
What would you expect to see in a museum of natural sciences? Frogs? Plants? Dinosaur skeletons? Exhibits about how the brain functions? A planetarium? Gems and minerals? Or, maybe all of the above? Science includes such diverse fields as astronomy, biology, computer sciences, geology, logic, physics, chemistry, and mathematics (Figure \(\PageIndex{3}\)). However, those fields of science related to the physical world and its phenomena and processes are considered natural sciences . Thus, a museum of natural sciences might contain any of the items listed above.
There is no complete agreement when it comes to defining what the natural sciences include, however. For some experts, the natural sciences are astronomy, biology, chemistry, earth science, and physics. Other scholars choose to divide natural sciences into life sciences , which study living things and include biology, and physical sciences , which study nonliving matter and include astronomy, geology, physics, and chemistry. Some disciplines such as biophysics and biochemistry build on both life and physical sciences and are interdisciplinary. Natural sciences are sometimes referred to as “hard science” because they rely on the use of quantitative data; social sciences that study society and human behavior are more likely to use qualitative assessments to drive investigations and findings.
Not surprisingly, the natural science of biology has many branches or subdisciplines. Cell biologists study cell structure and function, while biologists who study anatomy investigate the structure of an entire organism. Those biologists studying physiology, however, focus on the internal functioning of an organism. Some areas of biology focus on only particular types of living things. For example, botanists explore plants, while zoologists specialize in animals.
Scientific Reasoning
One thing is common to all forms of science: an ultimate goal “to know.” Curiosity and inquiry are the driving forces for the development of science. Scientists seek to understand the world and the way it operates. To do this, they use two methods of logical thinking: inductive reasoning and deductive reasoning.
Inductive reasoning is a form of logical thinking that uses related observations to arrive at a general conclusion. This type of reasoning is common in descriptive science. A life scientist such as a biologist makes observations and records them. These data can be qualitative or quantitative, and the raw data can be supplemented with drawings, pictures, photos, or videos. From many observations, the scientist can infer conclusions (inductions) based on evidence. Inductive reasoning involves formulating generalizations inferred from careful observation and the analysis of a large amount of data. Brain studies provide an example. In this type of research, many live brains are observed while people are doing a specific activity, such as viewing images of food. The part of the brain that “lights up” during this activity is then predicted to be the part controlling the response to the selected stimulus, in this case, images of food. The “lighting up” of the various areas of the brain is caused by excess absorption of radioactive sugar derivatives by active areas of the brain. The resultant increase in radioactivity is observed by a scanner. Then, researchers can stimulate that part of the brain to see if similar responses result.
Deductive reasoning or deduction is the type of logic used in hypothesis-based science. In deductive reason, the pattern of thinking moves in the opposite direction as compared to inductive reasoning. Deductive reasoning is a form of logical thinking that uses a general principle or law to forecast specific results. From those general principles, a scientist can extrapolate and predict the specific results that would be valid as long as the general principles are valid. Studies in climate change can illustrate this type of reasoning. For example, scientists may predict that if the climate becomes warmer in a particular region, then the distribution of plants and animals should change. These predictions have been made and tested, and many such changes have been found, such as the modification of arable areas for agriculture, with change based on temperature averages.
Both types of logical thinking are related to the two main pathways of scientific study: descriptive science and hypothesis-based science. Descriptive (or discovery) science , which is usually inductive, aims to observe, explore, and discover, while hypothesis-based science , which is usually deductive, begins with a specific question or problem and a potential answer or solution that can be tested. The boundary between these two forms of study is often blurred, and most scientific endeavors combine both approaches. The fuzzy boundary becomes apparent when thinking about how easily observation can lead to specific questions. For example, a gentleman in the 1940s observed that the burr seeds that stuck to his clothes and his dog’s fur had a tiny hook structure. On closer inspection, he discovered that the burrs’ gripping device was more reliable than a zipper. He eventually developed a company and produced the hook-and-loop fastener popularly known today as Velcro. Descriptive science and hypothesis-based science are in continuous dialogue.
The Scientific Method
Biologists study the living world by posing questions about it and seeking science-based responses. This approach is common to other sciences as well and is often referred to as the scientific method. The scientific method was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626) (Figure \(\PageIndex{4}\)), who set up inductive methods for scientific inquiry. The scientific method is not exclusively used by biologists but can be applied to almost all fields of study as a logical, rational problem-solving method.
The scientific process typically starts with an observation (often a problem to be solved) that leads to a question. Let’s think about a simple problem that starts with an observation and apply the scientific method to solve the problem. One Monday morning, a student arrives at class and quickly discovers that the classroom is too warm. That is an observation that also describes a problem: the classroom is too warm. The student then asks a question: “Why is the classroom so warm?”
Proposing a Hypothesis
Recall that a hypothesis is a suggested explanation that can be tested. To solve a problem, several hypotheses may be proposed. For example, one hypothesis might be, “The classroom is warm because no one turned on the air conditioning.” But there could be other responses to the question, and therefore other hypotheses may be proposed. A second hypothesis might be, “The classroom is warm because there is a power failure, and so the air conditioning doesn’t work.”
Once a hypothesis has been selected, the student can make a prediction. A prediction is similar to a hypothesis but it typically has the format “If . . . then . . . .” For example, the prediction for the first hypothesis might be, “ If the student turns on the air conditioning, then the classroom will no longer be too warm.”
Testing a Hypothesis
A valid hypothesis must be testable. It should also be falsifiable , meaning that it can be disproven by experimental results. Importantly, science does not claim to “prove” anything because scientific understandings are always subject to modification with further information. This step—openness to disproving ideas—is what distinguishes sciences from non-sciences. The presence of the supernatural, for instance, is neither testable nor falsifiable. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. The control group contains every feature of the experimental group except it is not given the manipulation that is hypothesized about. Therefore, if the results of the experimental group differ from the control group, the difference must be due to the hypothesized manipulation, rather than some outside factor. Look for the variables and controls in the examples that follow. To test the first hypothesis, the student would find out if the air conditioning is on. If the air conditioning is turned on but does not work, there should be another reason, and this hypothesis should be rejected. To test the second hypothesis, the student could check if the lights in the classroom are functional. If so, there is no power failure and this hypothesis should be rejected. Each hypothesis should be tested by carrying out appropriate experiments. Be aware that rejecting one hypothesis does not determine whether or not the other hypotheses can be accepted; it simply eliminates one hypothesis that is not valid (Figure \(\PageIndex{5}\)). Using the scientific method, the hypotheses that are inconsistent with experimental data are rejected.
While this “warm classroom” example is based on observational results, other hypotheses and experiments might have clearer controls. For instance, a student might attend class on Monday and realize she had difficulty concentrating on the lecture. One observation to explain this occurrence might be, “When I eat breakfast before class, I am better able to pay attention.” The student could then design an experiment with a control to test this hypothesis.
In hypothesis-based science, specific results are predicted from a general premise. This type of reasoning is called deductive reasoning: deduction proceeds from the general to the particular. But the reverse of the process is also possible: sometimes, scientists reach a general conclusion from a number of specific observations. This type of reasoning is called inductive reasoning, and it proceeds from the particular to the general. Inductive and deductive reasoning are often used in tandem to advance scientific knowledge (Figure \(\PageIndex{6}\)).
Art Connection
In the example below, the scientific method is used to solve an everyday problem. Order the scientific method steps (numbered items) with the process of solving the everyday problem (lettered items). Based on the results of the experiment, is the hypothesis correct? If it is incorrect, propose some alternative hypotheses.
- Observation
- Question
- Hypothesis (answer)
- Prediction
- Experiment
- Result
- There is something wrong with the electrical outlet.
- If something is wrong with the outlet, my coffeemaker also won’t work when plugged into it.
- My toaster doesn’t toast my bread.
- I plug my coffee maker into the outlet.
- My coffeemaker works.
- Why doesn’t my toaster work?
Art Connection
Decide if each of the following is an example of inductive or deductive reasoning.
- All flying birds and insects have wings. Birds and insects flap their wings as they move through the air. Therefore, wings enable flight.
- Insects generally survive mild winters better than harsh ones. Therefore, insect pests will become more problematic if global temperatures increase.
- Chromosomes, the carriers of DNA, separate into daughter cells during cell division. Therefore, DNA is the genetic material.
- Animals as diverse as humans, insects, and wolves all exhibit social behavior. Therefore, social behavior must have an evolutionary advantage.
The scientific method may seem too rigid and structured. It is important to keep in mind that, although scientists often follow this sequence, there is flexibility. Sometimes an experiment leads to conclusions that favor a change in approach; often, an experiment brings entirely new scientific questions to the puzzle. Many times, science does not operate in a linear fashion; instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests. Notice, too, that the scientific method can be applied to solving problems that aren’t necessarily scientific in nature.
Two Types of Science: Basic Science and Applied Science
The scientific community has been debating for the last few decades about the value of different types of science. Is it valuable to pursue science for the sake of simply gaining knowledge, or does scientific knowledge only have worth if we can apply it to solving a specific problem or to bettering our lives? This question focuses on the differences between two types of science: basic science and applied science.
Basic science or “pure” science seeks to expand knowledge regardless of the short-term application of that knowledge. It is not focused on developing a product or a service of immediate public or commercial value. The immediate goal of basic science is knowledge for knowledge’s sake, though this does not mean that, in the end, it may not result in a practical application.
In contrast, applied science or “technology,” aims to use science to solve real-world problems, making it possible, for example, to improve a crop yield, find a cure for a particular disease, or save animals threatened by a natural disaster (Figure \(\PageIndex{7}\)). In applied science, the problem is usually defined for the researcher.
Some individuals may perceive applied science as “useful” and basic science as “useless.” A question these people might pose to a scientist advocating knowledge acquisition would be, “What for?” A careful look at the history of science, however, reveals that basic knowledge has resulted in many remarkable applications of great value. Many scientists think that a basic understanding of science is necessary before an application is developed; therefore, applied science relies on the results generated through basic science. Other scientists think that it is time to move on from basic science and instead to find solutions to actual problems. Both approaches are valid. It is true that there are problems that demand immediate attention; however, few solutions would be found without the help of the wide knowledge foundation generated through basic science.
One example of how basic and applied science can work together to solve practical problems occurred after the discovery of DNA structure led to an understanding of the molecular mechanisms governing DNA replication. Strands of DNA, unique in every human, are found in our cells, where they provide the instructions necessary for life. During DNA replication, DNA makes new copies of itself, shortly before a cell divides. Understanding the mechanisms of DNA replication enabled scientists to develop laboratory techniques that are now used to identify genetic diseases, pinpoint individuals who were at a crime scene, and determine paternity. Without basic science, it is unlikely that applied science would exist.
Another example of the link between basic and applied research is the Human Genome Project, a study in which each human chromosome was analyzed and mapped to determine the precise sequence of DNA subunits and the exact location of each gene. (The gene is the basic unit of heredity; an individual’s complete collection of genes is his or her genome.) Other less complex organisms have also been studied as part of this project in order to gain a better understanding of human chromosomes. The Human Genome Project (Figure \(\PageIndex{8}\)) relied on basic research carried out with simple organisms and, later, with the human genome. An important end goal eventually became using the data for applied research, seeking cures and early diagnoses for genetically related diseases.
While research efforts in both basic science and applied science are usually carefully planned, it is important to note that some discoveries are made by serendipity , that is, by means of a fortunate accident or a lucky surprise. Penicillin was discovered when biologist Alexander Fleming accidentally left a petri dish of Staphylococcus bacteria open. An unwanted mold grew on the dish, killing the bacteria. The mold turned out to be Penicillium , and a new antibiotic was discovered. Even in the highly organized world of science, luck—when combined with an observant, curious mind—can lead to unexpected breakthroughs.
Reporting Scientific Work
Whether scientific research is basic science or applied science, scientists must share their findings in order for other researchers to expand and build upon their discoveries. Collaboration with other scientists—when planning, conducting, and analyzing results—are all important for scientific research. For this reason, important aspects of a scientist’s work are communicating with peers and disseminating results to peers. Scientists can share results by presenting them at a scientific meeting or conference, but this approach can reach only the select few who are present. Instead, most scientists present their results in peer-reviewed manuscripts that are published in scientific journals. Peer-reviewed manuscripts are scientific papers that are reviewed by a scientist’s colleagues, or peers. These colleagues are qualified individuals, often experts in the same research area, who judge whether or not the scientist’s work is suitable for publication. The process of peer review helps to ensure that the research described in a scientific paper or grant proposal is original, significant, logical, and thorough. Grant proposals, which are requests for research funding, are also subject to peer review. Scientists publish their work so other scientists can reproduce their experiments under similar or different conditions to expand on the findings. The experimental results must be consistent with the findings of other scientists.
A scientific paper is very different from creative writing. Although creativity is required to design experiments, there are fixed guidelines when it comes to presenting scientific results. First, scientific writing must be brief, concise, and accurate. A scientific paper needs to be succinct but detailed enough to allow peers to reproduce the experiments.
The scientific paper consists of several specific sections—introduction, materials and methods, results, and discussion. This structure is sometimes called the “IMRaD” format. There are usually acknowledgment and reference sections as well as an abstract (a concise summary) at the beginning of the paper. There might be additional sections depending on the type of paper and the journal where it will be published; for example, some review papers require an outline.
The introduction starts with brief, but broad, background information about what is known in the field. A good introduction also gives the rationale of the work; it justifies the work carried out and also briefly mentions the end of the paper, where the hypothesis or research question driving the research will be presented. The introduction refers to the published scientific work of others and therefore requires citations following the style of the journal. Using the work or ideas of others without proper citation is considered plagiarism .
The materials and methods section includes a complete and accurate description of the substances used, and the method and techniques used by the researchers to gather data. The description should be thorough enough to allow another researcher to repeat the experiment and obtain similar results, but it does not have to be verbose. This section will also include information on how measurements were made and what types of calculations and statistical analyses were used to examine raw data. Although the materials and methods section gives an accurate description of the experiments, it does not discuss them.
Some journals require a results section followed by a discussion section, but it is more common to combine both. If the journal does not allow the combination of both sections, the results section simply narrates the findings without any further interpretation. The results are presented by means of tables or graphs, but no duplicate information should be presented. In the discussion section, the researcher will interpret the results, describe how variables may be related, and attempt to explain the observations. It is indispensable to conduct an extensive literature search to put the results in the context of previously published scientific research. Therefore, proper citations are included in this section as well.
Finally, the conclusion section summarizes the importance of the experimental findings. While the scientific paper almost certainly answered one or more scientific questions that were stated, any good research should lead to more questions. Therefore, a well-done scientific paper leaves doors open for the researcher and others to continue and expand on the findings.
Review articles do not follow the IMRAD format because they do not present original scientific findings, or primary literature; instead, they summarize and comment on findings that were published as primary literature and typically include extensive reference sections.
Summary
Biology is the science that studies living organisms and their interactions with one another and their environments. Science attempts to describe and understand the nature of the universe in whole or in part by rational means. Science has many fields; those fields related to the physical world and its phenomena are considered natural sciences.
Science can be basic or applied. The main goal of basic science is to expand knowledge without any expectation of short-term practical application of that knowledge. The primary goal of applied research, however, is to solve practical problems.
Two types of logical reasoning are used in science. Inductive reasoning uses particular results to produce general scientific principles. Deductive reasoning is a form of logical thinking that predicts results by applying general principles. The common thread throughout scientific research is the use of the scientific method, a step-based process that consists of making observations, defining a problem, posing hypotheses, testing these hypotheses, and drawing one or more conclusions. The testing uses proper controls. Scientists present their results in peer-reviewed scientific papers published in scientific journals. A scientific research paper consists of several well-defined sections: introduction, materials and methods, results, and, finally, a concluding discussion. Review papers summarize the research done in a particular field over a period of time.
Art Connections
Figure \(\PageIndex{5}\): In the example below, the scientific method is used to solve an everyday problem. Order the scientific method steps (numbered items) with the process of solving the everyday problem (lettered items). Based on the results of the experiment, is the hypothesis correct? If it is incorrect, propose some alternative hypotheses.
- Observation
- Question
- Hypothesis (answer)
- Prediction
- Experiment
- Result
- There is something wrong with the electrical outlet.
- If something is wrong with the outlet, my coffeemaker also won’t work when plugged into it.
- My toaster doesn’t toast my bread.
- I plug my coffee maker into the outlet.
- My coffeemaker works.
- Why doesn’t my toaster work?
- Answer
-
1: C; 2: F; 3: A; 4: B; 5: D; 6: E. The original hypothesis is incorrect, as the coffeemaker works when plugged into the outlet. Alternative hypotheses include that the toaster might be broken or that the toaster wasn't turned on.
Figure \(\PageIndex{6}\): Decide if each of the following is an example of inductive or deductive reasoning.
- All flying birds and insects have wings. Birds and insects flap their wings as they move through the air. Therefore, wings enable flight.
- Insects generally survive mild winters better than harsh ones. Therefore, insect pests will become more problematic if global temperatures increase.
- Chromosomes, the carriers of DNA, separate into daughter cells during cell division. Therefore, DNA is the genetic material.
- Animals as diverse as humans, insects, and wolves all exhibit social behavior. Therefore, social behavior must have an evolutionary advantage.
- Answer
-
1: inductive; 2: deductive; 3: deductive; 4: inductive.
Glossary
- abstract
- opening section of a scientific paper that summarizes the research and conclusions
- applied science
- form of science that aims to solve real-world problems
- basic science
- science that seeks to expand knowledge and understanding regardless of the short-term application of that knowledge
- biology
- the study of living organisms and their interactions with one another and their environments
- conclusion
- section of a scientific paper that summarizes the importance of the experimental findings
- control
- part of an experiment that does not change during the experiment
- deductive reasoning
- form of logical thinking that uses a general inclusive statement to forecast specific results
- descriptive science
- (also, discovery science) form of science that aims to observe, explore, and investigate
- discussion
- section of a scientific paper in which the author interprets experimental results, describes how variables may be related, and attempts to explain the phenomenon in question
- falsifiable
- able to be disproven by experimental results
- hypothesis
- suggested explanation for an observation, which can be tested
- hypothesis-based science
- form of science that begins with a specific question and potential testable answers
- inductive reasoning
- form of logical thinking that uses related observations to arrive at a general conclusion
- introduction
- opening section of a scientific paper, which provides background information about what was known in the field prior to the research reported in the paper
- life science
- field of science, such as biology, that studies living things
- materials and methods
- section of a scientific paper that includes a complete description of the substances, methods, and techniques used by the researchers to gather data
- natural science
- field of science that is related to the physical world and its phenomena and processes
- peer-reviewed manuscript
- scientific paper that is reviewed by a scientist’s colleagues who are experts in the field of study
- physical science
- field of science, such as geology, astronomy, physics, and chemistry, that studies nonliving matter
- plagiarism
- using other people’s work or ideas without proper citation, creating the false impression that those are the author’s original ideas
- results
- section of a scientific paper in which the author narrates the experimental findings and presents relevant figures, pictures, diagrams, graphs, and tables, without any further interpretation
- review article
- paper that summarizes and comments on findings that were published as primary literature
- science
- knowledge that covers general truths or the operation of general laws, especially when acquired and tested by the scientific method
- scientific method
- method of research with defined steps that include observation, formulation of a hypothesis, testing, and confirming or falsifying the hypothesis
- serendipity
- fortunate accident or a lucky surprise
- theory
- tested and confirmed explanation for observations or phenomena
- variable
- part of an experiment that the experimenter can vary or change


1.4.7: Chemical Equilibria- Composition- Temperature and Pressure Dependence
The variable \(\xi\) describes in quite general terms molecular composition – molecular organisation. For a given closed system at fixed \(\mathrm{T}\) and \(p\) there exists a composition-organisation \(\xi_{\mathrm{eq}\) corresponding to a minimum in Gibbs energy where the affinity for spontaneous change is zero. In general terms there exists an extent of reaction \(\xi\) corresponding to a given affinity \(\mathrm{A}\) at defined \(\mathrm{T}\) and \(p\). In fact we can express \(\xi\) as a dependent variable defined by the independent variables, \(\mathrm{T}\), \(p\) and \(\mathrm{A}\).
\[\xi=\xi[T, p, A] \nonumber \]
The general differential of equation (a) takes the following form.
\[\mathrm{d} \xi=\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA} \nonumber \]
Equation (b) describes the dependence of extent of reaction on changes in \(\mathrm{T}\), \(p\) and affinity \(\mathrm{A}\).
\[\text { Moreover, }\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \nonumber \]
At equilibrium where ‘\(\mathrm{A} = 0\)’ and \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\), then
\[\left(\frac{\partial \xi}{\partial T}\right)_{p, A=0} \text { takes the sign of }\left[\frac{1}{T} \,\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}\right]=\left[\frac{1}{T} \, \sum_{j=1}^{j=i} v_{j} \, H_{j}^{e q}\right] \nonumber \]
\[\text { Similarly, }\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \text { takes the sign of }\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
Again at equilibrium where ‘\(\mathrm{A} = 0\)’ and \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\), then
\[\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \text { takes the sign of }\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}} \nonumber \]
Equations (d) and (f) are important being universally valid and forming the basis of important generalisations, the Laws of Moderation.
Equation (d) shows that the differential dependence of composition on temperature is related to the enthalpy of reaction. If the chemical reaction is exothermic {i.e. \(\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\) is negative}, the chemical equilibrium shifts to favour an increase in the amount of reactants. Whereas if the reaction is endothermic , the composition swings in a direction to favour the products.
In another experiment, the equilibrium system is perturbed by an increase in pressure. Equation (f) shows that the equilibrium composition swings to favour the reactants if the volume of reaction is positive. Alternatively if the volume of reaction is negative, the composition of the system changes to favour products [1].
Footnotes
[1] The conclusions reached here are called ‘Theorems of Moderation’. MJB was taught that the outcome is ‘Nature’s Laws of Cussedness’ [ = obstinacy]. An exothermic reaction generates heat to raise the temperature of the system, so the system responds, when the temperature is raised, by shifting the equilibrium in the direction for which the process is endothermic. The line of argument is not good thermodynamics but it makes the point.


Early learning is supported by attention to social-emotional development. Rather than taking time away from activities promoting learning and thinking, attention to the development of self, social interactions, and relationships is an essential component of an early childhood curriculum designed to promote learning in all young children. Here are some guiding principles on how to do that:
Support social-emotional development with intentionality and ample opportunities to practice skills
Attend to the impact of overall program design on social-emotional development (how you group children, what you model, etc.)
Utilize curriculum practices that support healthy social-emotional development, including:
allow many opportunities for practicing social interaction and relationship skills
provide support for the growth of age and developmentally appropriate self-regulation abilities
encourage curiosity and initiative
provide each child a network of nurturing, dependable adults who will actively support and scaffold his or her learning in a group setting
The most effective approach is play-based active learning
Here are some additional strategies to support children’s social and emotional development:
Create a program environment and daily routines that offer children opportunities for responsible and cooperative roles in the classroom or family child care community.
Model desirable behavior and attitudes in interactions with children and other adults.
Use the family culture to create bridges between the program and the home, supporting children’s pride in their family experience, and understand individual differences in background and viewpoint.
Enlist adults as active co-explorers in children’s chosen activities.
Encourage children’s ideas, initiative, and contributions to shared activities.
Observe children attentively, as they play, to understand each child’s needs, interests, strengths, and areas of growth in social-emotional development.
Establish developmentally and culturally appropriate expectations for children’s behavior, especially expectations for self-control and self-regulation.
Narrate for children what they are observed doing and expressing, providing language to describe their thoughts and feelings and to clarify others’ feelings.
Provide specific feedback to children about their efforts, reinforcing their choices that support learning and linking their actions to outcomes.
Coach and guide children’s behavior by using positive, respectful phrasing and tone to prompt problem solving and to give brief instructions and reminders.
Help children to understand social cues (facial expressions, body language, tone of voice). This can be fostered by simply allowing the children to freely play with their peers (learning through experience), or by modeling your own thought processes by thinking out loud (“I wonder what it means when Hayden is crying?”)
[1]


14.2: Light As a Particle
The only thing that interferes with my learning is my education. -- Albert Einstein
Radioactivity is random, but do the laws of physics exhibit randomness in other contexts besides radioactivity? Yes. Radioactive decay was just a good playpen to get us started with concepts of randomness, because all atoms of a given isotope are identical. By stocking the playpen with an unlimited supply of identical atom-toys, nature helped us to realize that their future behavior could be different regardless of their original identicality. We are now ready to leave the playpen, and see how randomness fits into the structure of physics at the most fundamental level.
The laws of physics describe light and matter, and the quantum revolution rewrote both descriptions. Radioactivity was a good example of matter's behaving in a way that was inconsistent with classical physics, but if we want to get under the hood and understand how nonclassical things happen, it will be easier to focus on light rather than matter. A radioactive atom such as uranium-235 is after all an extremely complex system, consisting of 92 protons, 143 neutrons, and 92 electrons. Light, however, can be a simple sine wave.
However successful the classical wave theory of light had been --- allowing the creation of radio and radar, for example --- it still failed to describe many important phenomena. An example that is currently of great interest is the way the ozone layer protects us from the dangerous short-wavelength ultraviolet part of the sun's spectrum. In the classical description, light is a wave. When a wave passes into and back out of a medium, its frequency is unchanged, and although its wavelength is altered while it is in the medium, it returns to its original value when the wave reemerges. Luckily for us, this is not at all what ultraviolet light does when it passes through the ozone layer, or the layer would offer no protection at all!
13.2.1 Evidence for light as a particle
For a long time, physicists tried to explain away the problems with the classical theory of light as arising from an imperfect understanding of atoms and the interaction of light with individual atoms and molecules. The ozone paradox, for example, could have been attributed to the incorrect assumption that one could think of the ozone layer as a smooth, continuous substance, when in reality it was made of individual ozone molecules. It wasn't until 1905 that Albert Einstein threw down the gauntlet, proposing that the problem had nothing to do with the details of light's interaction with atoms and everything to do with the fundamental nature of light itself.
In those days the data were sketchy, the ideas vague, and the experiments difficult to interpret; it took a genius like Einstein to cut through the thicket of confusion and find a simple solution. Today, however, we can get right to the heart of the matter with a piece of ordinary consumer electronics, the digital camera. Instead of film, a digital camera has a computer chip with its surface divided up into a grid of light-sensitive squares, called “pixels.” Compared to a grain of the silver compound used to make regular photographic film, a digital camera pixel is activated by an amount of light energy orders of magnitude smaller. We can learn something new about light by using a digital camera to detect smaller and smaller amounts of light, as shown in figure a . Figure a /1 is fake, but a /2 and a /3 are real digital-camera images made by Prof. Lyman Page of Princeton University as a classroom demonstration. Figure a /1 is what we would see if we used the digital camera to take a picture of a fairly dim source of light. In figures a /2 and a /3, the intensity of the light was drastically reduced by inserting semitransparent absorbers like the tinted plastic used in sunglasses. Going from a /1 to a /2 to a /3, more and more light energy is being thrown away by the absorbers.
The results are drastically different from what we would expect based on the wave theory of light. If light was a wave and nothing but a wave, b , then the absorbers would simply cut down the wave's amplitude across the whole wavefront. The digital camera's entire chip would be illuminated uniformly, and weakening the wave with an absorber would just mean that every pixel would take a long time to soak up enough energy to register a signal.
But figures a /2 and a /3 show that some pixels take strong hits while others pick up no energy at all. Instead of the wave picture, the image that is naturally evoked by the data is something more like a hail of bullets from a machine gun, c . Each “bullet” of light apparently carries only a tiny amount of energy, which is why detecting them individually requires a sensitive digital camera rather than an eye or a piece of film.
Although Einstein was interpreting different observations, this is the conclusion he reached in his 1905 paper: that the pure wave theory of light is an oversimplification, and that the energy of a beam of light comes in finite chunks rather than being spread smoothly throughout a region of space.
We now think of these chunks as particles of light, and call them “photons,” although Einstein avoided the word “particle,” and the word “photon” was invented later. Regardless of words, the trouble was that waves and particles seemed like inconsistent categories. The reaction to Einstein's paper could be kindly described as vigorously skeptical. Even twenty years later, Einstein wrote, “There are therefore now two theories of light, both indispensable, and --- as one must admit today despite twenty years of tremendous effort on the part of theoretical physicists --- without any logical connection.” In the remainder of this chapter we will learn how the seeming paradox was eventually resolved.
Discussion Questions
◊ Suppose someone rebuts the digital camera data in figure a , claiming that the random pattern of dots occurs not because of anything fundamental about the nature of light but simply because the camera's pixels are not all exactly the same --- some are just more sensitive than others. How could we test this interpretation?
◊ Discuss how the correspondence principle applies to the observations and concepts discussed in this section.
13.2.2 How much light is one photon?
The photoelectric effect
We have seen evidence that light energy comes in little chunks, so the next question to be asked is naturally how much energy is in one chunk. The most straightforward experimental avenue for addressing this question is a phenomenon known as the photoelectric effect. The photoelectric effect occurs when a photon strikes the surface of a solid object and knocks out an electron. It occurs continually all around you. It is happening right now at the surface of your skin and on the paper or computer screen from which you are reading these words. It does not ordinarily lead to any observable electrical effect, however, because on the average free electrons are wandering back in just as frequently as they are being ejected. (If an object did somehow lose a significant number of electrons, its growing net positive charge would begin attracting the electrons back more and more strongly.)
Figure e shows a practical method for detecting the photoelectric effect. Two very clean parallel metal plates (the electrodes of a capacitor) are sealed inside a vacuum tube, and only one plate is exposed to light. Because there is a good vacuum between the plates, any ejected electron that happens to be headed in the right direction will almost certainly reach the other capacitor plate without colliding with any air molecules.
The illuminated (bottom) plate is left with a net positive charge, and the unilluminated (top) plate acquires a negative charge from the electrons deposited on it. There is thus an electric field between the plates, and it is because of this field that the electrons' paths are curved, as shown in the diagram. However, since vacuum is a good insulator, any electrons that reach the top plate are prevented from responding to the electrical attraction by jumping back across the gap. Instead they are forced to make their way around the circuit, passing through an ammeter. The ammeter allows a measurement of the strength of the photoelectric effect.
An unexpected dependence on frequency
The photoelectric effect was discovered serendipitously by Heinrich Hertz in 1887, as he was experimenting with radio waves. He was not particularly interested in the phenomenon, but he did notice that the effect was produced strongly by ultraviolet light and more weakly by lower frequencies. Light whose frequency was lower than a certain critical value did not eject any electrons at all. (In fact this was all prior to Thomson's discovery of the electron, so Hertz would not have described the effect in terms of electrons --- we are discussing everything with the benefit of hindsight.) This dependence on frequency didn't make any sense in terms of the classical wave theory of light. A light wave consists of electric and magnetic fields. The stronger the fields, i.e., the greater the wave's amplitude, the greater the forces that would be exerted on electrons that found themselves bathed in the light. It should have been amplitude (brightness) that was relevant, not frequency. The dependence on frequency not only proves that the wave model of light needs modifying, but with the proper interpretation it allows us to determine how much energy is in one photon, and it also leads to a connection between the wave and particle models that we need in order to reconcile them.
To make any progress, we need to consider the physical process by which a photon would eject an electron from the metal electrode. A metal contains electrons that are free to move around. Ordinarily, in the interior of the metal, such an electron feels attractive forces from atoms in every direction around it. The forces cancel out. But if the electron happens to find itself at the surface of the metal, the attraction from the interior side is not balanced out by any attraction from outside. In popping out through the surface the electron therefore loses some amount of energy \(E_s\), which depends on the type of metal used.
Suppose a photon strikes an electron, annihilating itself and giving up all its energy to the electron. (We now know that this is what always happens in the photoelectric effect, although it had not yet been established in 1905 whether or not the photon was completely annihilated.) The electron will (1) lose kinetic energy through collisions with other electrons as it plows through the metal on its way to the surface; (2) lose an amount of kinetic energy equal to \(E_s\) as it emerges through the surface; and (3) lose more energy on its way across the gap between the plates, due to the electric field between the plates. Even if the electron happens to be right at the surface of the metal when it absorbs the photon, and even if the electric field between the plates has not yet built up very much, \(E_s\) is the bare minimum amount of energy that it must receive from the photon if it is to contribute to a measurable current. The reason for using very clean electrodes is to minimize \(E_s\) and make it have a definite value characteristic of the metal surface, not a mixture of values due to the various types of dirt and crud that are present in tiny amounts on all surfaces in everyday life.
We can now interpret the frequency dependence of the photoelectric effect in a simple way: apparently the amount of energy possessed by a photon is related to its frequency. A low-frequency red or infrared photon has an energy less than \(E_s\), so a beam of them will not produce any current. A high-frequency blue or violet photon, on the other hand, packs enough of a punch to allow an electron to make it to the other plate. At frequencies higher than the minimum, the photoelectric current continues to increase with the frequency of the light because of effects (1) and (3).
Numerical relationship between energy and frequency
Prompted by Einstein's photon paper, Robert Millikan (whom we first encountered in chapter 8 ) figured out how to use the photoelectric effect to probe precisely the link between frequency and photon energy. Rather than going into the historical details of Millikan's actual experiments (a lengthy experimental program that occupied a large part of his professional career) we will describe a simple version, shown in figure g , that is used sometimes in college laboratory courses. 2 The idea is simply to illuminate one plate of the vacuum tube with light of a single wavelength and monitor the voltage difference between the two plates as they charge up. Since the resistance of a voltmeter is very high (much higher than the resistance of an ammeter), we can assume to a good approximation that electrons reaching the top plate are stuck there permanently, so the voltage will keep on increasing for as long as electrons are making it across the vacuum tube.
At a moment when the voltage difference has a reached a value \(\Delta \)V, the minimum energy required by an electron to make it out of the bottom plate and across the gap to the other plate is \(E_s+e\Delta \)V. As \(\Delta V\) increases, we eventually reach a point at which \(E_s+e\Delta V\) equals the energy of one photon. No more electrons can cross the gap, and the reading on the voltmeter stops rising. The quantity \(E_s+e\Delta V\) now tells us the energy of one photon. If we determine this energy for a variety of wavelengths, h , we find the following simple relationship between the energy of a photon and the frequency of the light:
where \(h\) is a constant with the value \(6.63\times10^{-34}\ \text{J}\cdot\text{s}\). Note how the equation brings the wave and particle models of light under the same roof: the left side is the energy of one particle of light, while the right side is the frequency of the same light, interpreted as a wave . The constant \(h\) is known as Planck's constant, for historical reasons explained in the footnote beginning on the preceding page.
self-check:
How would you extract \(h\) from the graph in figure h ? What if you didn't even know \(E_s\) in advance, and could only graph \(e\Delta V\) versus \(f\)?
(answer in the back of the PDF version of the book)
Since the energy of a photon is \(hf\), a beam of light can only have energies of \(hf\), \(2hf\), \(3hf\), etc. Its energy is quantized --- there is no such thing as a fraction of a photon. Quantum physics gets its name from the fact that it quantizes quantities like energy, momentum, and angular momentum that had previously been thought to be smooth, continuous and infinitely divisible.
| Example 7: Number of photons emitted by a lightbulb per second |
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\(\triangleright\) Roughly how many photons are emitted by a 100-W lightbulb in 1 second? \(\triangleright\) People tend to remember wavelengths rather than frequencies for visible light. The bulb emits photons with a range of frequencies and wavelengths, but let's take 600 nm as a typical wavelength for purposes of estimation. The energy of a single photon is
\[\begin{align*} E_{photon} &= hf \\ &= hc/\lambda \end{align*}\]
A power of 100 W means 100 joules per second, so the number of photons is
\[\begin{align*} (100\ \text{J})/E_{photon} &= (100\ \text{J}) / (hc/\lambda ) \\ &\approx 3\times10^{20} \end{align*}\]
This hugeness of this number is consistent with the correspondence principle. The experiments that established the classical theory of optics weren't wrong. They were right, within their domain of applicability, in which the number of photons was so large as to be indistinguishable from a continuous beam. |
| Example 8: Measuring the wave |
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When surfers are out on the water waiting for their chance to catch a wave, they're interested in both the height of the waves and when the waves are going to arrive. In other words, they observe both the amplitude and phase of the waves, and it doesn't matter to them that the water is granular at the molecular level. The correspondence principle requires that we be able to do the same thing for electromagnetic waves, since the classical theory of electricity and magnetism was all stated and verified experimentally in terms of the fields \(\mathbf{E}\) and \(\mathbf{B}\), which are the amplitude of an electromagnetic wave. The phase is also necessary, since the induction effects predicted by Maxwell's equation would flip their signs depending on whether an oscillating field is on its way up or on its way back down. This is a more demanding application of the correspondence principle than the one in example 7 , since amplitudes and phases constitute more detailed information than the over-all intensity of a beam of light. Eyeball measurements can't detect this type of information, since the eye is much bigger than a wavelength, but for example an AM radio receiver can do it with radio waves, since the wavelength for a station at 1000 kHz is about 300 meters, which is much larger than the antenna. The correspondence principle demands that we be able to explain this in terms of the photon theory, and this requires not just that we have a large number of photons emitted by the transmitter per second, as in example 7 , but that even by the time they spread out and reach the receiving antenna, there should be many photons overlapping each other within a space of one cubic wavelength. Problem 47 on p. 903 verifies that the number is in fact extremely large. |
| Example 9: Momentum of a photon |
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\(\triangleright\) According to the theory of relativity, the momentum of a beam of light is given by \(p=E/c\). Apply this to find the momentum of a single photon in terms of its frequency, and in terms of its wavelength. \(\triangleright\) Combining the equations \(p=E/c\) and \(E=hf\), we find
\[\begin{align*} p &= E/c \\ &= \frac{h}{c}f . \end{align*}\]
To reexpress this in terms of wavelength, we use \(c=f\lambda \):
\[\begin{align*} p &= \frac{h}{c}\cdot\frac{c}{\lambda} \\ &= \frac{h}{\lambda} \end{align*}\]
The second form turns out to be simpler. |
Discussion Questions
◊ The photoelectric effect only ever ejects a very tiny percentage of the electrons available near the surface of an object. How well does this agree with the wave model of light, and how well with the particle model? Consider the two different distance scales involved: the wavelength of the light, and the size of an atom, which is on the order of \(10^{-10}\) or \(10^{-9}\) m.
◊ What is the significance of the fact that Planck's constant is numerically very small? How would our everyday experience of light be different if it was not so small?
◊ How would the experiments described above be affected if a single electron was likely to get hit by more than one photon?
◊ Draw some representative trajectories of electrons for \(\Delta V=0\), \(\Delta V\) less than the maximum value, and \(\Delta V\) greater than the maximum value.
◊ Explain based on the photon theory of light why ultraviolet light would be more likely than visible or infrared light to cause cancer by damaging DNA molecules. How does this relate to discussion question C?
◊ Does \(E=hf\) imply that a photon changes its energy when it passes from one transparent material into another substance with a different index of refraction?
13.2.3 Wave-particle duality
How can light be both a particle and a wave? We are now ready to resolve this seeming contradiction. Often in science when something seems paradoxical, it's because we (1) don't define our terms carefully, or (2) don't test our ideas against any specific real-world situation. Let's define particles and waves as follows:
- Waves exhibit superposition, and specifically interference phenomena.
- Particles can only exist in whole numbers, not fractions.
As a real-world check on our philosophizing, there is one particular experiment that works perfectly. We set up a double-slit interference experiment that we know will produce a diffraction pattern if light is an honest-to-goodness wave, but we detect the light with a detector that is capable of sensing individual photons, e.g., a digital camera. To make it possible to pick out individual dots due to individual photons, we must use filters to cut down the intensity of the light to a very low level, just as in the photos by Prof. Page on p. 837. The whole thing is sealed inside a light-tight box. The results are shown in figure i . (In fact, the similar figures in on page 837 are simply cutouts from these figures.)
Neither the pure wave theory nor the pure particle theory can explain the results. If light was only a particle and not a wave, there would be no interference effect. The result of the experiment would be like firing a hail of bullets through a double slit, j . Only two spots directly behind the slits would be hit.
If, on the other hand, light was only a wave and not a particle, we would get the same kind of diffraction pattern that would happen with a water wave, k . There would be no discrete dots in the photo, only a diffraction pattern that shaded smoothly between light and dark.
Applying the definitions to this experiment, light must be both a particle and a wave. It is a wave because it exhibits interference effects. At the same time, the fact that the photographs contain discrete dots is a direct demonstration that light refuses to be split into units of less than a single photon. There can only be whole numbers of photons: four photons in figure i /3, for example.
A wrong interpretation: photons interfering with each other
One possible interpretation of wave-particle duality that occurred to physicists early in the game was that perhaps the interference effects came from photons interacting with each other. By analogy, a water wave consists of moving water molecules, and interference of water waves results ultimately from all the mutual pushes and pulls of the molecules. This interpretation was conclusively disproved by G.I. Taylor, a student at Cambridge. The demonstration by Prof. Page that we've just been discussing is essentially a modernized version of Taylor's work. Taylor reasoned that if interference effects came from photons interacting with each other, a bare minimum of two photons would have to be present at the same time to produce interference. By making the light source extremely dim, we can be virtually certain that there are never two photons in the box at the same time. In figure i /3, however, the intensity of the light has been cut down so much by the absorbers that if it was in the open, the average separation between photons would be on the order of a kilometer! At any given moment, the number of photons in the box is most likely to be zero. It is virtually certain that there were never two photons in the box at once.
The concept of a photon's path is undefined.
If a single photon can demonstrate double-slit interference, then which slit did it pass through? The unavoidable answer must be that it passes through both! This might not seem so strange if we think of the photon as a wave, but it is highly counterintuitive if we try to visualize it as a particle. The moral is that we should not think in terms of the path of a photon. Like the fully human and fully divine Jesus of Christian theology, a photon is supposed to be 100% wave and 100% particle. If a photon had a well defined path, then it would not demonstrate wave superposition and interference effects, contradicting its wave nature. (In subsection 13.3.4 we will discuss the Heisenberg uncertainty principle, which gives a numerical way of approaching this issue.)
Another wrong interpretation: the pilot wave hypothesis
A second possible explanation of wave-particle duality was taken seriously in the early history of quantum mechanics. What if the photon particle is like a surfer riding on top of its accompanying wave ? As the wave travels along, the particle is pushed, or “piloted” by it. Imagining the particle and the wave as two separate entities allows us to avoid the seemingly paradoxical idea that a photon is both at once. The wave happily does its wave tricks, like superposition and interference, and the particle acts like a respectable particle, resolutely refusing to be in two different places at once. If the wave, for instance, undergoes destructive interference, becoming nearly zero in a particular region of space, then the particle simply is not guided into that region.
The problem with the pilot wave interpretation is that the only way it can be experimentally tested or verified is if someone manages to detach the particle from the wave, and show that there really are two entities involved, not just one. Part of the scientific method is that hypotheses are supposed to be experimentally testable. Since nobody has ever managed to separate the wavelike part of a photon from the particle part, the interpretation is not useful or meaningful in a scientific sense.
The probability interpretation
The correct interpretation of wave-particle duality is suggested by the random nature of the experiment we've been discussing: even though every photon wave/particle is prepared and released in the same way, the location at which it is eventually detected by the digital camera is different every time. The idea of the probability interpretation of wave-particle duality is that the location of the photon-particle is random, but the probability that it is in a certain location is higher where the photon-wave's amplitude is greater.
More specifically, the probability distribution of the particle must be proportional to the square of the wave's amplitude,
This follows from the correspondence principle and from the fact that a wave's energy density is proportional to the square of its amplitude. If we run the double-slit experiment for a long enough time, the pattern of dots fills in and becomes very smooth as would have been expected in classical physics. To preserve the correspondence between classical and quantum physics, the amount of energy deposited in a given region of the picture over the long run must be proportional to the square of the wave's amplitude. The amount of energy deposited in a certain area depends on the number of photons picked up, which is proportional to the probability of finding any given photon there.
| Example 10: A microwave oven |
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\(\triangleright\) The figure shows two-dimensional (top) and one-dimensional (bottom) representations of the standing wave inside a microwave oven. Gray represents zero field, and white and black signify the strongest fields, with white being a field that is in the opposite direction compared to black. Compare the probabilities of detecting a microwave photon at points A, B, and C.
\(\triangleright\) A and C are both extremes of the wave, so the probabilities of detecting a photon at A and C are equal. It doesn't matter that we have represented C as negative and A as positive, because it is the square of the amplitude that is relevant. The amplitude at B is about 1/2 as much as the others, so the probability of detecting a photon there is about 1/4 as much. |
The probability interpretation was disturbing to physicists who had spent their previous careers working in the deterministic world of classical physics, and ironically the most strenuous objections against it were raised by Einstein, who had invented the photon concept in the first place. The probability interpretation has nevertheless passed every experimental test, and is now as well established as any part of physics.
An aspect of the probability interpretation that has made many people uneasy is that the process of detecting and recording the photon's position seems to have a magical ability to get rid of the wavelike side of the photon's personality and force it to decide for once and for all where it really wants to be. But detection or measurement is after all only a physical process like any other, governed by the same laws of physics. We will postpone a detailed discussion of this issue until p. 864, since a measuring device like a digital camera is made of matter, but we have so far only discussed how quantum mechanics relates to light.
| Example 11: What is the proportionality constant? |
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\(\triangleright\) What is the proportionality constant that would make an actual equation out of \((\text{probability distribution})\propto(\text{amplitude})^2\)? \(\triangleright\) The probability that the photon is in a certain small region of volume \(v\) should equal the fraction of the wave's energy that is within that volume. For a sinusoidal wave, which has a single, well-defined frequency \(f\), this gives
\[\begin{align*} P &= \frac{\text{energy in volume $v$}}{\text{energy of photon}} \\ &= \frac{\text{energy in volume $v$}}{hf} . \end{align*}\]
We assume \(v\) is small enough so that the electric and magnetic fields are nearly constant throughout it. We then have
\[\begin{equation*} P = \frac{\left(\frac{1}{8\pi k}|\mathbf{E}|^2 +\frac{c^2}{8\pi k}|\mathbf{B}|^2\right)v}{hf} . \end{equation*}\]
We can simplify this formidable looking expression by recognizing that in a plane wave, \(|\mathbf{E}|\) and \(|\mathbf{B}|\) are related by \(|\mathbf{E}|=c|\mathbf{B}|\). This implies (problem 40 , p. 725), that the electric and magnetic fields each contribute half the total energy, so we can simplify the result to
\[\begin{align*} P &= 2\frac{\left(\frac{1}{8\pi k}|\mathbf{E}|^2\right)v}{hf} \\ &= \frac{v}{4\pi khf}|\mathbf{E}|^2 . \end{align*}\]
The probability is proportional to the square of the wave's amplitude, as advertised. 3 |
Discussion Questions
◊ Referring back to the example of the carrot in the microwave oven, show that it would be nonsensical to have probability be proportional to the field itself, rather than the square of the field.
◊ Einstein did not try to reconcile the wave and particle theories of light, and did not say much about their apparent inconsistency. Einstein basically visualized a beam of light as a stream of bullets coming from a machine gun. In the photoelectric effect, a photon “bullet” would only hit one atom, just as a real bullet would only hit one person. Suppose someone reading his 1905 paper wanted to interpret it by saying that Einstein's so-called particles of light are simply short wave-trains that only occupy a small region of space. Comparing the wavelength of visible light (a few hundred nm) to the size of an atom (on the order of 0.1 nm), explain why this poses a difficulty for reconciling the particle and wave theories.
◊ Can a white photon exist?
◊ In double-slit diffraction of photons, would you get the same pattern of dots on the digital camera image if you covered one slit? Why should it matter whether you give the photon two choices or only one?
13.2.4 Photons in three dimensions
Up until now I've been sneaky and avoided a full discussion of the three-dimensional aspects of the probability interpretation. The example of the carrot in the microwave oven, for example, reduced to a one-dimensional situation because we were considering three points along the same line and because we were only comparing ratios of probabilities. The purpose of bringing it up now is to head off any feeling that you've been cheated conceptually rather than to prepare you for mathematical problem solving in three dimensions, which would not be appropriate for the level of this course.
A typical example of a probability distribution in section 13.1 was the distribution of heights of human beings. The thing that varied randomly, height, \(h\), had units of meters, and the probability distribution was a graph of a function \(D(h)\). The units of the probability distribution had to be \(\text{m}^{-1}\) (inverse meters) so that areas under the curve, interpreted as probabilities, would be unitless: \((\text{area})=(\text{height})(\text{width})=\text{m}^{-1}\cdot\text{m}\).
Now suppose we have a two-dimensional problem, e.g., the probability distribution for the place on the surface of a digital camera chip where a photon will be detected. The point where it is detected would be described with two variables, \(x\) and \(y\), each having units of meters. The probability distribution will be a function of both variables, \(D(x,y)\). A probability is now visualized as the volume under the surface described by the function \(D(x,y)\), as shown in figure n . The units of \(D\) must be \(\text{m}^{-2}\) so that probabilities will be unitless: \((\text{probability})=(\text{depth})(\text{length})(\text{width}) =\text{m}^{-2}\cdot\text{m}\cdot\text{m}\). In terms of calculus, we have \(P\:=\:\int Ddx dy\).
Generalizing finally to three dimensions, we find by analogy that the probability distribution will be a function of all three coordinates, \(D(x,y,z)\), and will have units of \(\text{m}^{-3}\). It is unfortunately impossible to visualize the graph unless you are a mutant with a natural feel for life in four dimensions. If the probability distribution is nearly constant within a certain volume of space \(v\), the probability that the photon is in that volume is simply \(vD\). If not, then we can use an integral, \(P\:=\:\int Ddx dydz\).
Contributors
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


2.5: Head-to-Toe Assessment
A comprehensive head-to-toe assessment is done on patient admission, at the beginning of each shift, and when it is determined to be necessary by the patient’s hemodynamic status and the context. The head-to-toe assessment includes all the body systems, and the findings will inform the health care professional on the patient’s overall condition. Any unusual findings should be followed up with a focused assessment specific to the affected body system.
A physical examination involves collecting objective data using the techniques of inspection, palpation, percussion, and auscultation as appropriate (Wilson & Giddens, 2013). Checklist 17 outlines the steps to take.
Checklist 17: Head-to-Toe AssessmentDisclaimer: Always review and follow your hospital policy regarding this specific skill. |
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Safety considerations:
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Steps |
Additional Information |
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1. General appearance:
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Alterations may reflect neurologic impairment, oral injury or impairment, improperly fitting dentures, differences in dialect or language, or potential mental illness. Unusual findings should be followed up with a focused neurological system assessment. | ||
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This is not a specific step. Evaluating the skin, hair, and nails is an ongoing element of a full body assessment as you work through steps 3-9.
2. Skin, hair, and nails:
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Check for and follow up on the presence of lesions, bruising, and rashes.Variations in skin temperature, texture, and perspiration or dehydration may indicate underlying conditions.
Redness of the skin at pressure areas such as heels, elbows, buttocks, and hips indicates the need to reassess patient’s need for position changes. Unilateral edema may indicate a local or peripheral cause, whereas bilateral-pitting edema usually indicates cardiac or kidney failure. Check hair for the presence of lice and/or nits (eggs), which are oval in shape and adhere to the hair shaft. |
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3. Head and neck:
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Check eyes for drainage, pupil size, and reaction to light. Drainage may indicate infection, allergy, or injury.
Slow pupillary reaction to light or unequal reactions bilaterally may indicate neurological impairment. Dry mucous membranes indicate decreased hydration. Facial asymmetry may indicate neurological impairment or injury. Unusual findings should be followed up with a focused neurological system assessment. |
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4. Chest:
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Chest expansion may be asymmetrical with conditions such as atelectasis, pneumonia, fractured ribs, or pneumothorax.
Use of accessory muscles may indicate acute airway obstruction or massive atelectasis. Jugular distension of more than 3 cm above the sternal angle while the patient is at 45º may indicate cardiac failure. The presence of crackles or wheezing must be further assessed, documented, and reported. Unusual findings should be followed up with a focused respiratory assessment. Note the heart rate and rhythm, identify S1 and S2, and follow up on any unusual findings with a focused cardiovascular assessment. |
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5. Abdomen:
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Abdominal distension may indicate ascites associated with conditions such as heart failure, cirrhosis, and pancreatitis. Markedly visible peristalsis with abdominal distension may indicate intestinal obstruction.
Hyperactive bowel sounds may indicate bowel obstruction, gastroenteritis, or subsiding paralytic ileum. Hypoactive or absent bowel sounds may be present after abdominal surgery, or with peritonitis or paralytic ileus. Pain and tenderness may indicate underlying inflammatory conditions such as peritonitis. Unusual findings in urine output may indicate compromised urinary function. Follow up with a focused gastrointestinal and genitourinary assessment. Unusual findings with bowel movements should be followed up with a focused gastrointestinal and genitourinary assessment. |
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6. Extremities:
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Limitation in range of movement may indicate articular disease or injury.
Palpate pulses for symmetry in rate and rhythm. Asymmetry may indicate cardiovascular conditions or post-surgical complications. Unequal handgrip and/or foot strength may indicate underlying conditions, injury, or post-surgical complications. CWMS : colour, warmth, movement, and sensation of the hands and feet should be checked and compared to determine adequacy of perfusion. Check skin integrity and pressure areas, and ensure follow-up and in-depth assessment of patient mobility and need for regular changes in position. Palpate and inspect capillary refill and report if more than 3 seconds. To check capillary refill, depress the nail edge to cause blanching and then release. Colour should return to the nail instantly or in less than 3 seconds. If it takes longer, this suggests decreased peripheral perfusion and may indicate cardiovascular or respiratory dysfunction. Unusual findings should be followed up with a focused cardiovascular assessment. Clubbing of nails, in which the nails present as straightened out to 180 degrees, with the nail base feeling spongy, occurs with heart disease, emphysema, and chronic bronchitis. |
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7. Back area (turn patient to side or ask to sit up or lean forward):
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Check for curvature or abnormalities in the spine.
Check skin integrity and pressure areas, and ensure follow-up and in-depth assessment of patient mobility and need for regular changes in position. |
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8. Tubes, drains, dressings, and IVs:
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Note amount, colour, and consistency of drainage (e.g., Foley catheter), or if infusing as prescribed (e.g., intravenous).
Assess wounds for large amounts of drainage or for purulent drainage, and provide wound care as indicated. |
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9. Mobility:
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Assess patient’s risk for falls. Document and follow up any indication of falls risk. Note use of mobility aids and ensure they are available to the patient on ambulation. | ||
| 10. Report and document assessment findings and related health problems according to agency policy. | Accurate and timely documentation and reporting promote patient safety. | ||
| Data source: Assessment Skill Checklists, 2014; Jarvis et al., 2014; Stephen et al., 2012 |
Critical Thinking Exercises
- You are assessing a patient at the beginning of your shift. Which assessment would be the most appropriate?
- You come back from a break to find your patient complaining that she feels short of breath. Which assessment would be the most appropriate?