Wave Function

For hundreds of years, the scientific community perceived physics as the most precise science capable of plotting the motion of the planets and explaining the thermodynamics needed to power engines and machines. Who would have foreseen that Max Planck's innocent 1900 proposal that radiation consists of discrete energy packets would completely overturn humanity's presumed understanding of microphysics? Indeed, the dawn of quantum mechanics was quite a shock as physicists began to discover that nature was far from precise. In fact, probability and uncertainty were fundamental features of nature and not just reflections of our ignorance! At the heart of this probabilistic theory is the wave function which gives the most complete description of a quantum system possible. In this article, we will look deeper at the wave function and how we can visualize quantum systems. In doing so we'll come across one of the most intriguing and controversial questions in modern physics, the infamous collapse of the wave function.

Get started

Millions of flashcards designed to help you ace your studies

Sign up for free

Review generated flashcards

Sign up for free
You have reached the daily AI limit

Start learning or create your own AI flashcards

StudySmarter Editorial Team

Team Wave Function Teachers

  • 12 minutes reading time
  • Checked by StudySmarter Editorial Team
Save Article Save Article
Contents
Contents

Jump to a key chapter

    Wave Function Definition

    Quantum mechanics defines the state of a system probabilistically. This means that one cannot know the precise state of a system before making a measurement. Mathematically, a quantum wave function denoted by \(\Psi\) encodes these probabilities. This quantum wave function is a function of the degrees of freedom defining the possible states of the system. The wave function then outputs a complex number, known as the probability amplitude, whose modulus squared gives the probability density of the system being in that particular state.

    Complex numbers are numbers with both a real and an imaginary component. They are of the form \(x+yi\), with \(i\) defined by \(i^2=-1\). The modulus squared of a complex number, \(z=x+iy\), is defined as \[|z|^2=zz^*=(x+iy)(x-iy)=x^2+y^2.\]

    The wave function contains all the information about a system and how it evolves over time. However, the laws of quantum mechanics restrict our experimental access to this information. Born's rule, one of the fundamental postulates of quantum mechanics, describes the relationship between the wave function and the probability density associated with the likelihood of measuring the system to be in some state.

    Born's rule states that the probability density of finding a particle in a particular state is proportional to the modulus squared of the wave function at that point.

    Mathematically, one would write this as

    \[\text{Probability to be found at}\, x \propto\, |\Psi(x)|^2.\]

    Much like regular waves, wave functions can interfere. For example, in the double-slit experiment, the wave function of an electron passing through one slit interferes with the wave function of the same electron passing through the other slit. This is where the complex value of the wave function comes into play because the wave functions can have different complex phases between them. When two wave functions interfere with each other their complex amplitudes add together, either causing constructive interference or destructive interference.

    Consider two wave functions whose amplitude at \(x\) is

    \[\Psi_1(x)=A,\,\Psi_2(x)=B.\]

    When these two wavefunctions interfere, the resulting amplitude of the combined wavefunction will be

    \[\Psi_{12}=A+B\]

    with an overall probability of

    \[|\Psi_{12}|^2=|\Psi_{1}+\Psi_{2}|^2=|A+B|^2.\]

    For complex numbers \(A\) and \(B\) this is not the same as simply adding the two probabilities separately. In general, the following inequality holds instead:

    \[|A+B|^2\leq|A|^2+|B|^2.\]

    The above inequality, which appears across many fields of research, is called the triangle inequality. Any time we want to establish a distance measure in an arbitrary geometrical space, we have to appeal to it. Because it's so general, mathematicians regard the triangle inequality as a fundamental result.

    Electron Wave Function

    One of the first great advancements in quantum physics came when Erwin Schrödinger applied the concept of a quantum wave function to electrons in an atom. A quantum wave function describes the position of an electron around the nucleus within an atom. The wave function is not a precise orbit, but rather a 'probability cloud' with regions where it is more or less likely to find the electron.

    Wave Function Graphic showing the wave function of an electron in a hydrogen atom at various energy levels StudySmarterFig. 1. Image showing the probability densities of an electron around a nucleus in a hydrogen atom. The lighter the color the more likely the electron will be found there. This probability density is defined by the wavefunction \(\Psi\).

    The energy level or orbital of the electron within the atom determines the shape of the probability cloud. Figure one shows the orbitals of an electron within a hydrogen atom. The electron is most likely to be found in the lighter regions whilst it will never be found in the black regions. In addition, the so-called quantum numbers characterize the shape of the orbital by defining the angular momentum and spin of the electron. At low energies, the orbital has a ring-like shape similar to the Bohr Model of circular orbits. However, at higher energies, the shape of these orbitals becomes more complex. This wave function model for atomic electrons is the most accurate picture of the atom we have and has been key to developing our theory of chemical bonding. Consequently, one of quantum mechanics' most important applications is its prediction of the properties of the periodic table.

    Wave Function Graph

    We can visualize the behavior of a quantum particle with the graph of its wave function. By looking at the graph's amplitude, we can see which regions are most likely to contain the particle. Charting how the graph changes over time also indicates how the quantum particle evolves. For example, look at the graph below representing the wave function of a particle in a box. The wave function is a standing wave with four nodes and three anti-nodes. The nodes of the particle's wave function are the regions where the probability of finding the particle are lowest, whereas the anti-nodes are regions where the probability to find the particle is highest.

    Remember it is \(|\Psi(x)|^2\) that gives the probability of measuring the particle at the position \(x\). Negative values of the wave function do not represent low probabilities, they simply represent a phase difference.

    Wave Functions Graph of a wave function of a particle trapped in a well with positions of highest and zero probability annotated StudySmarterFig.2- A graph of the wave function of a particle trapped in a box. The standing wave formed shows that there are zones of high probability and areas of zero probability, much like the electron orbitals in an atom.

    We can see this probability distribution more directly by graphing the absolute value squared of the wave function

    \[|\Psi(x)|^2\]

    The figure below shows the result. In this figure, the three clear peaks correspond to the anti-nodes of the standing wave.

    Wave Function Graph of the absolute value squared of the standing wave function StudySmarterFig. 3. Plotting \(|\Psi(x)|^2\) gives the probability density distribution of the particles position.

    The graph of \(|\Psi(x)|^2\) is a continuous probability distribution giving the probability density of the particle at each point. The area under the graph in some region \({a<x<b}\) gives the probability of finding the particle. This is equivalent to the usual procedure of summing probabilities in a discrete probability distribution. A larger area means that the probability of finding the particle there is higher. As we would expect, increasing the size of the region within which we look for the particle means a larger area under the graph and a greater chance of finding the particle.

    Normalization of the Wave Function

    We can place some key constraints on a wave function to ensure it describes physical particles in agreement with experiments. Consider searching for a particle within a box. Obviously, the particle must be somewhere if we look over the entire region of the box. Mathematically, we say that the sum over the probabilities for each position must always be \(1\). As we saw in the previous section, the area under the graph of \(|\Psi(x)|^2\) within some region is equivalent to this sum over probabilities. It follows then that the total area under the \(|\Psi(x)|^2\) graph must always be one. Note that this applies to any wave function. Even if \(\Psi(x)\) extends over an infinite space (as it does for free particles), the area under the \(|\Psi(x)|^2\) graph must always equal one.

    Wave Function Graph of the absolute value squared of the standing wave function with the enclosed area shaded and the area equal to one StudySmarter

    Fig.4 - The wavefunction \(\Psi(x)\) of a particle in a box is normalized such that the area under the \(|\Psi(x)|^2\) graph is equal to one.

    This may seem like a massive constraint on the number of physically permissible wave functions. However, it is possible to find a physically permissible wave function from almost any wave function by multiplying it by a suitable factor, known as a normalization factor. Consider a graph of \(|\Psi(x)|^2\) , denoted \(A\) for some wave function \(\Psi(x)\). If the area under the curve is not equal to one, we can multiply \(\Psi(x)\) by \(\frac{1}{A}\) to ensure that the new wave function is physical.\[\Psi_N(x)=\frac{1}{A}\Psi(x).\]

    Such a process is called normalizing the wave function \(\Psi(x)\). Incredibly, it turns out that the physics described by the normalized wave function \(\Psi_N(x)\) is entirely equivalent to the original un-normalized wave function. However, this normalization procedure only works for wave functions where \(0<A<\infty\).

    Wave Function Collapse

    It is important to keep in mind that the wave function of a system determines the probability of finding the system in some state or position upon measurement. However, something strange happens once we do make that measurement. Let's say we wanted to check the results of our first measurement by making a second measurement shortly after. For the result of our first measurement to be valid, this second measurement must return the same result. Otherwise, there is nothing to confirm our initial measurement. This means that once measured, a quantum system must remain in the same state. This has a profound implication for our understanding of the wave function.

    Consider measuring the position of a particle, again described by some wave function. Initially, the wave function is a spread of probability-amplitudes over space. However, once we make a measurement we know for certain the position of the particle. Let's call this position \(C\). Any further measurement must always return \(C\), thereby making the probability at \(C\) one and zero everywhere else. This means the wave function becomes a sharp peak centered at \(C\) with zero amplitude everywhere else.

    Wave Function A graph of the wavefunction of a particle after measurement shown as a sharp peak demonstrating wave function collapse StudySmarterFig. 5. After measuring the particle within the box, its wave function collapses down to a single point shown above as a sharp peak around \(x=1\). The probability of finding the particle at \(x=1\) is now \(1\).

    We say that the measurement has 'collapsed' the wave function down to a single point. This collapse of the wave function demonstrates the mysterious nature of measurements in quantum mechanics. It is one of the strangest and most controversial aspects of quantum physics. The physics behind such a collapse, and whether such a physical interpretation is even possible, is still a topic of fierce debate.

    Asking what happens during wave function collapse is such a difficult conceptual problem that many physicists have given up on trying to solve it altogether. Instead, they adopt a "shut up and calculate" mentality in which they trust the ability of the mathematics of quantum mechanics to make successful predictions without worrying too deeply about the meaning of the theory. Philosophers of physics, on the other hand, do care significantly about establishing the meaning of quantum mechanics. By studying the historical development and logical foundations of quantum mechanics, they hope to solve this enigma once and for all.

    Wave Function - Key takeaways

    • Quantum systems are described by a complex wave function that defines a probability distribution over the possible states of the system.
    • The amplitude of a wave function at a point \(x\), \(\Psi(x)\), is a complex number whose absolute value squared gives the probability of finding the system in the state \(x\)\[\text{Probability of system in x}=|\Psi(x)|^2\]
    • The quantum description of electrons in an atom is one where electrons exist within 'probability clouds' around the atom defined by the wave function of the electron at a particular orbital.
    • If the wave function for a particle's position is plotted graphically, the particle is most likely to be found at positions corresponding to the extrema of the wave function.
    • The area under a wave function within some interval defines the probability of the particle being found within that interval.

    References

    1. Fig.1 - Hydrogen density plots (https://commons.wikimedia.org/wiki/File:Hydrogen_Density_Plots.png) by PoorLeno is under Public Domain.
    2. Fig.2 - Annotated wave function graph, StudySmarter Originals.
    3. Fig.3-Wave function modulus squared graph, StudySmarter Originals.
    4. Fig.4-Area under a normalized wave function, StudySmarter Originals.
    5. Fig.5- Graph of wave function collapse, StudySmarter Originals
    Frequently Asked Questions about Wave Function

    What does the wave function ψ represent? 

    The wave function represents the probability of measuring a system to be in some state.

    What is a wave function?

    A wave function in quantum mechanics is a mathematical function that describes the state of a quantum system. Its output is a complex probability-amplitude describing the state of the system, whose modulus squared is the probability of the system to be in that state when measured.

    Why does a wave-function have to be normalized?

    As the modulus square of the wave-function is a probability, it must only take values between 0 and 1, else it would violate the laws of probabilities. In order to ensure this holds, the wave function must be multiplied by an overall normalization factor.

    What is wave function with example?

    A wave function in quantum mechanics is a mathematical function that describes the state of a quantum system. For example the wave function of a free moving particle is described using a Gaussian wave function. 

    Save Article

    Test your knowledge with multiple choice flashcards

    The probability of finding a particle in a particular position is given by the ___.

    The output of the wave function is what kind of number?

    If two wave functions with amplitudes \(\Psi_1(x)=A\) and \(\Psi_2(x)=B\) interfere with each other, the probability associated with the combined wave function is always given by \(\Psi_{12}=|A|^2+|B|^2\).

    Next

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Physics Teachers

    • 12 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App
    Sign up with Email