Quantum Physics For Dummies
Overview
The plain-English guide to understanding quantum physics
Mastering quantum physics is no easy feat, but with the help of Quantum Physics For Dummies you can work at your own pace to unlock key concepts and fascinating facts. Packed with invaluable explanations, equations, and step-by-step instructions, this book makes a challenging subject much more accessible. Great for college students taking a quantum physics course, Quantum Physics For Dummies offers complete coverage of the subject, along with numerous examples to help you tackle the tough stuff. The Schrodinger Equation, the foundations of quantum physics, vector notation, scattering theory, angular momentum—it’s all in here. This handy guide helps you prepare for exams and succeed at learning quantum physics.
- Get clear explanations of the core concepts in quantum physics
- Review the math principles needed for quantum physics equations
- Learn the latest breakthroughs and research in the field
- Clarify difficult subjects and equations from your college course
Quantum Physics For Dummies is great a resource for students who need a supplement to the textbook to help them tackle this challenging subject.
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About The Author
Andrew Zimmerman Jones is a researcher, educator, and science writer. He is the former physics guide at About.com, where he wrote lessons and explanations for physics questions. Andrew earned his degree in physics from Wabash College and his master’s degree in mathematics education from Purdue University. He is the author of String Theory For Dummies.
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quantum physics for dummies
CHEAT SHEET
This Cheat Sheet is intended to supplement Quantum Physics For Dummies, 3rd edition, by Andrew Zimmerman Jones. It begins by reviewing some useful operators used in quantum mechanics calculations. Then it covers a useful method for solving the Schrödinger equation for the quantum wave function, and then how you can use that wave function to calculate probabilities in quantum physics.
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Quantum Physics
At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution forif you remember that, for a free particle,That equation gives you this form forBecausethe equation turns intoIn fact, now that the right side of the equation is in terms of the radius vector r, you can make the left side match:That’s the solution to the Schrödinger equation, but it’s unphysical.
Quantum Physics
In quantum physics, you can apply the radial equation inside a square well (where the radius is greater than zero and less than a). For a spherical square well potential, here's what the radial equation looks like for the region 0 r a:
In this region, V(r) = –V0, so you have
Taking the V0 term over to the right gives you the following:
And here's what dividing by r gives you:
Then, multiplying by
you get
Now make the change of variable
Using this substitution means that
This is the spherical Bessel equation.
Quantum Physics
In quantum physics, you can apply the radial equation outside a square well (where the radius is greater than a). In the region r > a, the particle is just like a free particle, so here's what the radial equation looks like:
You solve this equation as follows:
you substitute
so that Rnl(r) becomes
Using this substitution means that the radial equation takes the following form:
The solution is a combination of spherical Bessel functions and spherical Neumann functions, where Bl is a constant:
If the energy E l = i Bl", so that the wave function decays exponentially at large distances r.
Quantum Physics
In quantum physics, you can apply the Schrödinger equation when you work on problems that have a central potential. These are problems where you're able to separate the wave function into a radial part (which depends on the form of the potential) and an angular part, which is a spherical harmonic.
Central potentials are spherically symmetrical potentials, of the kind where V(r) = V(r).
Quantum Physics
In quantum physics, you can apply the spherical Bessel and Neumann functions to a free particle (a particle which is not constrained by any potential). The wave function in spherical coordinates takes this form:
and
gives you the spherical harmonics. The problem is now to solve for the radial part, Rnl(r). Here's the radial equation:
For a free particle, V(r) = 0, so the radial equation becomes
The way you usually handle this equation is to substitute
and because you have a version of the same equation for each n index it is convenient to simply remove it, so that Rnl (r) becomes
This substitution means that
becomes the following:
The radial part of the equation looks tough, but the solutions turn out to be well-known — this equation is called the spherical Bessel equation, and the solution is a combination of the spherical Bessel functions
and the spherical Neumann functions
where Al and Bl are constants.
Quantum Physics
When you want to find where an electron is at any given time in a hydrogen atom, what you’re actually doing is finding how far the electron is from the proton. You can find the expectation value of r, that is, , to tell you its location. Given that the wave function is
the following expression represents the probability that the electron will be found in the spatial element d3r:
In spherical coordinates,
So you can write
as
The probability that the electron is in a spherical shell of radius r to r + dr is therefore
And because
this equation becomes the following:
The preceding equation is equal to
(Remember that the asterisk symbol [*] means the complex conjugate.
Quantum Physics
If your quantum physics instructor asks you to find the wave function of a hydrogen atom, you can start with the radial Schrödinger equation, Rnl(r), which tells you that
The preceding equation comes from solving the radial Schrödinger equation:
The solution is only good to a multiplicative constant, so you add such a constant, Anl (which turns out to depend on the principal quantum number n and the angular momentum quantum number l), like this:
You find Anl by normalizing Rnl(r).
Quantum Physics
When you apply the quantum mechanical Schrödinger equation for a hydrogen atom, you need to put together the solutions for small r and large r. The Schrödinger equation gives you a solution to the radial Schrödinger equation for a hydrogen atom as follows:
where f(r) is some as-yet-undetermined function of r. Your next task is to determine f(r), which you do by substituting this equation into the radial Schrödinger equation, giving you the following:
Performing the substitution gives you the following differential equation:
Quite a differential equation, eh?
Quantum Physics
In quantum physics, many of the wave functions that are solutions to physical setups like the square well aren’t inherently symmetric or antisymmetric; they’re simply asymmetric. In other words, they have no definite symmetry. So how do you end up with symmetric or antisymmetric wave functions?
The answer is that you have to create them yourself, and you do that by adding together asymmetric wave functions.
Quantum Physics
In quantum physics, you can put together the symmetric and antisymmetric wave functions of a system of three or more particles from single-particle wave functions. The symmetric wave function looks like this:
And the antisymmetric wave function looks like this:
This asymmetric wave function goes to zero if any two single particles have the same set of quantum numbers
How about generalizing this to systems of N particles?
Quantum Physics
If your quantum physics instructor asks you to create symmetric and antisymmetric wave functions for a two-particle system, you can start with the single-particle wave functions:
By analogy, here’s the symmetric wave function, made up of two single-particle wave functions:
And here’s the antisymmetric wave function, made up of the two single-particle wave functions:
where ni stands for all the quantum numbers of the ith particle.
Quantum Physics
Here’s an example that involves finding the rotational energy spectrum of a diatomic molecule. The figure shows the setup: A rotating diatomic molecule is composed of two atoms with masses m1 and m2. The first atom rotates at r = r1, and the second atom rotates at r = r2. What’s the molecule’s rotational energy?
Quantum Physics
In quantum physics, you can determine the angular part of a wave function when you work on problems that have a central potential. With central potential problems, you're able to separate the wave function into an angular part, which is a spherical harmonic, and a radial part (which depends on the form of the potential).
Quantum Physics
In quantum physics, to be able to determine the energy levels of a particle in a box potential, you need an exact value for X(x) — not just one of the terms of the constants A and B. You have to use the boundary conditions to find A and B. What are the boundary conditions? The wave function must disappear at the boundaries of the box, so
X(0) = 0
X(Lx) = 0
So the fact that
tells you right away that B must be 0, because cos(0) = 1.
Quantum Physics
In quantum physics, you can determine the radial part of a wave function when you work on problems that have a central potential. With central potential problems, you’re able to separate the wave function into a radial part (which depends on the form of the potential) and an angular part, which is a spherical harmonic.
In analogy with orbital angular momentum, you can assume that m (the z-axis component of spin) can take the values –s, –s + 1, ..., s – 1, and s, where s is the total spin quantum number. For electrons, physicists Otto Stern and Walther Gerlach observed two spots, so you have 2s + 1 = 2, which means that s = 1/2.
Quantum Physics
At some point, your quantum physics instructor may ask you to find the eigenfunctions of Lz in spherical coordinates. In spherical coordinates, the Lz operator looks like this:
which is the following:
And because
this equation can be written in this version:
Cancelling out terms from the two sides of this equation gives you this differential equation:
This looks easy to solve, and the solution is just
where C is a constant of integration.
Quantum Physics
In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s z component of angular momentum.
Start by taking a look at L+, and plan to solve for c:
L+| l, m > = c | l, m + 1 >
So L+ | l, m > gives you a new state, and multiplying that new state by its transpose should give you c2:
To see this equation, note that
On the other hand, also note that
so you have
What do you do about L+ L–?
Quantum Physics
The Stern-Gerlach experiment unexpectedly revealed the existence of spin back in 1922. Physicists Otto Stern and Walther Gerlach sent a beam of silver atoms through the poles of a magnet — whose magnetic field was in the z direction — as you can see in the following figure.
The Stern-Gerlach experiment.
Because 46 of silver’s 47 electrons are arranged in a symmetrical cloud, they contribute nothing to the orbital angular momentum of the atom.
Quantum Physics
In quantum physics, you can use the Schrödinger equation to see how the wave function for a particle in an infinite square well evolves with time. The Schrödinger equation looks like this:
You can also write the Schrödinger equation this way, where H is the Hermitian Hamiltonian operator:
That’s actually the time-independent Schrödinger equation.
Quantum Physics
In quantum physics, you can find the limits for small and large rho of a free particle. According to the spherical Bessel equation, the radial part of the wave function for a free particle looks like this:
Take a look at the spherical Bessel functions and Neumann functions for
Note that the Neumann functions diverge for
Therefore, any wave function that includes the Neumann functions also diverges, which is unphysical.
Quantum Physics
Using the Schrödinger equation tells you just about all you need to know about the hydrogen atom, and it's all based on a single assumption: that the wave function must go to zero as r goes to infinity, which is what makes solving the Schrödinger equation possible.
Hydrogen atoms are composed of a single proton, around which revolves a single electron.
Quantum Physics
At some point, your quantum physics instructor may want you to find the total energy equation for three-dimensional free particle problems. The total energy of the free particle is the sum of the energy in three dimensions:
E = Ex + Ey + Ez
With a free particle, the energy of the x component of the wave function is
And this equation works the same way for the y and z components, so here’s the total energy of the particle:
Note that kx2 + ky2 + kz2 is the square of the magnitude of k — that is,
Therefore, you can write the equation for the total energy as
Note that because E is a constant, no matter where the particle is pointed, all the eigenfunctions of
are infinitely degenerate as you vary kx, ky, and kz.
Quantum Physics
At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,You can write its general solution aswhere Ay and Az are constants.Becauseyou get this forwhere A= Ax Ay Az.The part i
Quantum Physics
By observing both pair production and pair annihilation, 20th-century physicists were able to prove that light has the characteristics of a particle. This process of discovery began in 1928, when the physicist Paul Dirac posited the existence of a positively charged anti-electron, the positron. He did this by taking the newly evolving field of quantum physics to new territory by combining relativity with quantum mechanics to create relativistic quantum mechanics — and that was the theory that predicted, through a plus/minus–sign interchange — the existence of the positron.
Quantum Physics
When you are working with potential barrier of height V0 and width a where E > V0, this means that the particle has enough energy to pass through the potential barrier and end up in the x > a region. This is what the Schrödinger equation looks like in this case:
The solutions for
are the following:
In fact, because there's no leftward traveling wave in the x > a region,
So how do you determine A, B, C, D, and F?
Quantum Physics
When a particle doesn't have as much energy as the potential of a barrier, you can use the Schrödinger equation to find the probability that the particle will tunnel through the barrier's potential. You can also find the reflection and transmission coefficients, R and T, as well as calculate the transmission coefficient using the Wentzel-Kramers-Brillouin (WKB) approximation.
Quantum Physics
The photoelectric effect was one of many experimental results that made up a crisis for classical physics around the turn of the 20th century. It was also one of Einstein’s first successes, and it provides proof of the quantization of light. Here’s what happened.
The photoelectric effect.
When you shine light onto metal, as the figure above shows, you get emitted electrons.
Quantum Physics
One of the major ideas of quantum physics is quantization — measuring quantities in discrete, not continuous, units. The idea of quantized energies arose with one of the earliest challenges to classical physics: the problem of black-body radiation. While Wien’s formula and the Rayleigh-Jeans Law could not explain the spectrum of a black body, Max Planck’s equation solved the problem by assuming that light was discrete.
Quantum Physics
Quantum physics, unlike classical physics, is completely nondeterministic. You can never know the precise position and momentum of a particle at any one time. You can give only probabilities for these linked measurements.
In quantum physics, the state of a particle is described by a wave function,
The wave function describes the de Broglie wave of a particle, giving its amplitude as a function of position and time.
Quantum Physics
Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instructor will tell you, there are analogous spin operators, S2 and Sz, to orbital angular momentum operators L2 and Lz. However, these operators are just operators; they don’t have a differential form like the orbital angular momentum operators do.
Quantum Physics
In a hydrogen atom, the wave functions change as you change the orbital radius, r. So what do the hydrogen wave functions look like? Given that
looks like this:
Here are some other hydrogen wave functions:
Note that
behaves like rl for small r and therefore goes to zero. And for large r,
decays exponentially to zero.
Quantum Physics
In 1923, the physicist Louis de Broglie suggested that not only did waves exhibit particle-like aspects, but that the reverse was also true — all material particles should display wave-like properties.
How does this work? For a photon, momentum
And the wave vector, k, is equal to
De Broglie presented these apparently surprising suggestions in his Ph.
Quantum Physics
Although Max Planck and Albert Einstein postulated that light could behave as both a wave and a particle, it was Arthur Compton who finally proved that this was possible. His experiment involved scattering photons off electrons, as the figure below shows, and offered proof for what we now refer to as the Compton effect.
Quantum Physics
A multi-electron atom is the most common multi-particle system that quantum physics considers. You can apply a Hamiltonian wave function to a neutral, multi-electron atom, as shown in the following figure. Here, R is the coordinate of the nucleus (relative to the center of mass), r1 is the coordinate of the first electron (relative to the center of mass), r2 the coordinate of the second electron, and so on.
In quantum physics, the eigenvectors of a Hermitian operator define a complete set of orthonormal vectors — that is, a complete basis for the state space. When viewed in this “eigenbasis,” which is built of the eigenvectors (note that eigen is German for “innate” or “natural”), the operator in matrix format is diagonal and the elements along the diagonal of the matrix are the eigenvalues.
Quantum Physics
In quantum physics, probabilities take the place of absolute measurements. Say you've been experimenting with rolling a pair of dice and are trying to figure the relative probability that the dice will show various values. You come up with a list indicating the relative probability of rolling a 2, 3, 4, and so on, all the way up to 12:
Sum of the Dice
Relative Probability (Number of Ways of Rolling a Particular
Total)
2
1
3
2
4
3
5
4
6
5
7
6
8
5
9
4
10
3
11
2
12
1
In other words, you're twice as likely to roll a 3 than a 2, you're four times as likely to roll a 5 than a 2, and so on.
Quantum Physics
Each quantum state of the hydrogen atom is specified with three quantum numbers: n (the principal quantum number), l (the angular momentum quantum number of the electron), and m (the z component of the electron’s angular momentum,How many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers n, l, and m?
Quantum Physics
When calculating the spin of an electron in a hydrogen atom, you need to allow for the spin of the electron, which provides additional quantum states. Given the following equation, where the wave function of the hydrogen atom is a product of radial and angular parts,
you can add a spin part, corresponding to the spin of the electron, where s is the spin of the electron and ms is the z component of the spin:
The spin part of the equation can take the following values:
Hence,
now becomes
And this wave function can take two different forms, depending on ms, like this:
In fact, you can use the spin notation, where
For example, for
you can write the wave function as
And for
you can write the wave function as
What does this do to the energy degeneracy?
Quantum Physics
In quantum physics, to find the actual eigenfunctions (not just the eigenstates) of angular momentum operators like L2 and Lz, you turn from rectangular coordinates, x, y, and z, to spherical coordinates because it’ll make the math much simpler (after all, angular momentum is about things going around in circles).
Quantum Physics
You can determine what happens to the wave function when you swap particles in a multi-particle atom. Whether the wave function is symmetric or antisymmetric under such operations gives you insight into whether two particles can occupy the same quantum state.
Given that Pij2 = 1, note that if a wave function is an eigenfunction of Pij, then the possible eigenvalues are 1 and –1.
You can create the actual eigenstates, | l, m >, of angular momentum states in quantum mechanics. When you have the eigenstates, you also have the eigenvalues, and when you have the eigenvalues, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum.
Don't make the assumption that the eigenstates are | l, m >; rather, say they're
where the eigenvalue of
So the eigenvalue of
Similarly, the eigenvalue of
To proceed further, you have to introduce raising and lowering operators.
Quantum Physics
In quantum physics, you can decouple systems of particles that you can distinguish — that is, systems of identifiably different particles — into linearly independent equations. To illustrate this, suppose you have a system of many different types of cars floating around in space. You can distinguish all those cars because they’re all different — they have different masses, for one thing.
Quantum Physics
From a time-independent quantum physics point of view, you can derive the incident wave and scattered wave functions of two spinless nonrelativistic particles. To do so, you need to assume that the interaction between the particles depends only on their relative distance, |r1 – r2|.
You can reduce problems of this kind to two decoupled problems.
In quantum physics, the Schrödinger technique, which involves wave mechanics, uses wave functions, mostly in the position basis, to reduce questions in quantum physics to a differential equation.
Werner Heisenberg developed the matrix-oriented view of quantum physics, sometimes called matrix mechanics. The matrix representation is fine for many problems, but sometimes you have to go past it, as you’re about to see.
Quantum Physics
In quantum physics, when you have the eigenstates of a system, you can determine the allowable states of the system and the relative probability that the system will be in any of those states.
The commutator of operators A, B is [A, B] = AB – BA, so note that the commutator of
is the following:
This is equal to the following:
This equation breaks down to
And putting together this equation with the Hamiltonian,
Okay, with the commutator relations, you’re ready to go.
Quantum Physics
When you apply the quantum mechanical Schrödinger equation for a hydrogen atom, the quantization condition for the wave function of r to remain finite as r goes to infinity is
where
Substituting
into the quantization-condition equation gives you the following:
Now solve for the energy, E. Squaring both sides of the preceding equation gives you
So here’s the energy, E (Note: Because E depends on the principal quantum number, you rename it En):
Physicists often write this result in terms of the Bohr radius — the orbital radius that Niels Bohr calculated for the electron in a hydrogen atom, r0.
Quantum Physics
Quantum mechanically, identical particles in a multi-particle system don’t retain their individuality in terms of any measurable, observable quantity. You lose the individuality of identical particles as soon as you mix them with similar particles. This idea holds true for any N-particle system. As soon as you let N identical particles interact, you can’t say which exact one is at r1 or r2 or r3 or r4 and so on.
Quantum Physics
If you have a number of solutions to the Schrödinger equation, any linear combination of those solutions is also a solution. So that’s the key to getting a physical particle: You add various wave functions together so that you get a wave packet, which is a collection of wave functions of the form
such that the wave functions interfere constructively at one location and interfere destructively (go to zero) at all other locations:
This is usually written as a continuous integral:
What is
It’s the amplitude of each component wave function, and you can find
from the Fourier transform of the equation:
Because
you can also write the wave packet equations like this, in terms of p, not k:
Well, you may be asking yourself just what’s going on here.
When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum. The eigenvalues of the angular momentum are the possible values the angular momentum can take.
Here’s how to derive eigenstate equations with
Note that L2 – Lz2 = Lx2 + Ly2, which is a positive number, so
That means that
And substituting in
and using the fact that the eigenstates are normalized, gives you this:
So there’s a maximum possible value of
which you can call
You can be clever now, because there has to be a state
such that you can’t raise
any more.
Quantum Physics
If you can determine the wave function for the ground state of a quantum mechanical harmonic oscillator, then you can find any excited state of that harmonic oscillator.
So, if you know what
looks like, you can determine the first excited state,
Say you’re given this as your starting point:
And you know that
is the following:
And because
you get the following equation:
Given that the wave function for the ground state of a quantum mechanical harmonic oscillator is
So what does
look like?
Quantum Physics
In quantum physics, you can find commutators of angular momentum, L. First examine Lx, Ly, and Lz by taking a look at how they commute; if they commute (for example, if [Lx, Ly] = 0), then you can measure any two of them (Lx and Ly, for example) exactly. If not, then they’re subject to the uncertainty relation, and you can’t measure them simultaneously exactly.
In quantum physics, if you know the boundary conditions of a square well, you can find theenergy levels of an electron.
The equation
tells you that you have to use the boundary conditions to find the constants A and B. What are the boundary conditions? The wave function must disappear at the boundaries of an infinite square well, so
The fact that
tells you right away that B must be zero because cos(0) = 1.
Quantum Physics
Infinite square well, in which the walls go to infinity, is a favorite problem in quantum physics. To solve for the wave function of a particle trapped in an infinite square well, you can simply solve the Schrödinger equation.
Take a look at the infinite square well in the figure.
A square well.
Here’s what that square well looks like:
The Schrödinger equation looks like this in three dimensions:
Writing out the Schrödinger equation gives you the following:
You’re interested in only one dimension — x (distance) — in this instance, so the Schrödinger equation looks like
Because V(x) = 0 inside the well, the equation becomes
And in problems of this sort, the equation is usually written as
So now you have a second-order differential equation to solve for the wave function of a particle trapped in an infinite square well.
In quantum physics, the measure of how different it is to apply operator A and then B, versus B and then A, is called the operators’ commutator. Here’s how you define the commutator of operators A and B:
Two operators commute with each other if their commutator is equal to zero. That is, it doesn’t make any difference in what order you apply them:
Note in particular that any operator commutes with itself:
And it’s easy to show that the commutator of A, B is the negative of the commutator of B, A:
It’s also true that commutators are linear— that is,
And the Hermitian adjoint of a commutator works this way:
You can also find the anticommutator, {A, B}:
Here’s another one: What can you say about the Hermitian adjoint of the commutator of two Hermitian operators?
Quantum Physics
Your quantum physics instructor may ask you to find the eigenfunctions of L2 in spherical coordinates. To do this, you start with the eigenfunction of
given that in spherical coordinates, the L2 operator looks like this:
That’s quite an operator. And, given that
you can apply the L2 operator to
which gives you the following:
And because
this equation becomes
Wow, what have you gotten into?
Quantum Physics
Using quantum physics, you can determine the f eigenvalues and matching eigenvectors for systems in which the energies are degenerate. Take a look at this unperturbed Hamiltonian:
In other words, several states have the same energy. Say the energy states are f-fold degenerate, like this:
How does this affect the perturbation picture?
Quantum Physics
In quantum physics, if you’re given an operator in matrix form, you can find its eigenvectors and eigenvalues. For example, say you need to solve the following equation:
First, you can rewrite this equation as the following:
I represents the identity matrix, with 1s along its diagonal and 0s otherwise:
Remember that the solution to
exists only if the determinant of the matrix A – aI is 0:
det(A – aI) = 0
How to find the eigenvalues
Any values of a that satisfy the equation det(A – aI) = 0 are eigenvalues of the original equation.
Quantum Physics
In quantum physics, you can use operators to determine the energy eigenstate of a harmonic oscillator in position space. The charm of using the operators a and
is that given the ground state, | 0 >, those operators let you find all successive energy states. If you want to find an excited state of a harmonic oscillator, you can start with the ground state, | 0 >, and apply the raising operator,
For example, you can do this:
And so on.
Quantum Physics
Your quantum physics instructor may ask you to find the energy level of a harmonic oscillator. The best way to learn how is through an example. Say that you have a proton undergoing harmonic oscillation with
as shown in the figure.
A proton undergoing harmonic oscillation.
What are the energies of the various energy levels of the proton?
Quantum Physics
In quantum physics, when you have the exact eigenvalues for a charged oscillator in a perturbed system, you can find the energy of the system. Based on the perturbation theory, the corrected energy of the oscillator is given by
where
is the perturbation term in the Hamiltonian. That is, here,
Now take a look at the corrected energy equation using
The first-order correction is
which, using
becomes
because that's the expectation value of x, and harmonic oscillators spend as much time in negative x territory as in positive x territory — that is, the average value of x is zero.
Quantum Physics
In quantum physics, in order to find the energy of a perturbed system, En, you need to start by calculating the energy and wave function of an unperturbed system. You start with the energy:
Then add the first-order correction to the energy,
and add the second-order correction to the energy,
Now what about the wave function of the perturbed system,
Start with the wave function of the unperturbed system,
Add to it the first-order correction,
And then add to that the second-order correction to the wave function,
Note that when
becomes the unperturbed energy:
and
becomes the unperturbed wave function:
So your task is to calculate E(1)n and E(2)n, as well as
So how do you do that in general?
Quantum Physics
Given that everything in quantum physics is done in terms of probabilities, making predictions becomes very important. And the biggest such prediction is the expectation value. The expectation value of an operator is the average value that you would measure if you performed the measurement many times. For example, the expectation value of the Hamiltonian operator is the average energy of the system you’re studying.
Quantum Physics
In quantum physics, in order to find the first-order corrections to energy levels and wave functions of a perturbed system, En, you need to calculate E(1)n, as well as
So how do you do that? You start with three perturbed equations:
You then combine these three equations to get this jumbo equation:
You can handle the jumbo equation by setting the coefficients of lambda on either side of the equal sign to each other.
Quantum Physics
If you’ve read through the last few sections, you’re now armed with all this new technology: Hermitian operators and commutators. How can you put it to work? You can come up with the Heisenberg uncertainty relation starting virtually from scratch.
Here’s a calculation that takes you from a few basic definitions to the Heisenberg uncertainty relation.
In quantum physics, you’ll often work with Hermitian adjoints. The Hermitian adjoint — also called the adjoint or Hermitian conjugate — of an operator A is denoted
To find the Hermitian adjoint, you follow these steps:
Replace complex constants with their complex conjugates.
The Hermitian adjoint of a complex number is the complex conjugate of that number:
Replace kets with their corresponding bras, and replace bras with their corresponding kets.
Finding the inverse of a large matrix often isn’t easy, so quantum physics calculations are sometimes limited to working with unitary operators, U, where the operator’s inverse is equal to its adjoint,
(To find the adjoint of an operator, A, you find the transpose by interchanging the rows and columns, AT. Then take the complex conjugate,
Note that the asterisk (*) symbol in the above equation means the complex conjugate.
Quantum Physics
In quantum physics, if you are given the wave equation for a particle in an infinite square well, you may be asked to normalize the wave function. For example, start with the following wave equation:
The wave function is a sine wave, going to zero at x = 0 and x = a. You can see the first two wave functions plotted in the following figure.
Quantum Physics
Say you have two particles of equal mass colliding in a lab frame (where one particle starts at rest). You want to show that the two particles end up traveling at right angles with respect to each other in the lab frame. Using quantum physics, you can.
Note that if m1 = m2, then
Note also that
You know that
tells you that the following is true:
So substituting
into the preceding equation gives you
Therefore,
the angles of the particles in the lab frame after the collision, add up to
are at right angles with respect to each other.
Quantum Physics
In quantum physics, in order to find the second-order corrections to energy levels and wave functions of a perturbed system, En, you need to calculate E(2)n, as well as
So how do you do that? You start with three perturbed equations:
You then combine these three equations to get this jumbo equation:
From the jumbo equation, you can then find the second-order corrections to the energy levels and the wave functions.
Quantum Physics
The Hamiltonian represents the total energy of all the particles in a multi-particle system. You can describe that system in quantum physics terms. The following figure shows a multi-particle system where a number of particles are identified by their position (ignoring spin).
To find the total energy for this system, begin by working with the wave function.
Quantum Physics
In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.
The ground state of a quantum mechanical harmonic oscillator.
As a gaussian curve, the ground state of a quantum oscillator is
How can you figure out A?
Quantum Physics
In quantum physics, when finding the solution for a radial equation for a hydrogen atom, you need to keep the function of r finite as r approaches infinity to prevent the solution from becoming unphysical. You can accomplish this by putting constraints on the allowable values of the energy, and causing the solution for the radial equation to go to zero as r goes to infinity.
Quantum Physics
In your quantum physics course, you may be asked to normalize the wave function in a box potential. Here's an example: consider the wave function
In the x dimension, you have this for the wave equation:
So the wave function is a sine wave, going to zero at x = 0 and x = Lz. You can also insist that the wave function be normalized, like this:
By normalizing the wave function, you can solve for the unknown constant A.
Quantum Physics
Quantum physics experiments take place in the lab frame, but you do scattering calculations in the center-of-mass frame, so you have to know how to relate the angle between the two frames.
Here's how this works: The following figure shows scattering in the lab frame.
Scattering in the lab frame.
One particle, traveling at
is incident on another particle that's at rest
and hits it.
Quantum Physics
The scattering amplitude of spinless particles is crucial to understanding scattering from the quantum physics point of view. To see that, take a look at the current densities, Jinc (the flux density of a given incident particle) and Jsc (the current density for a given scattered particle):
(Remember that the asterisk symbol [*] means the complex conjugate.
Quantum Physics
Infinite square wells, in which the walls go to infinity, are a favorite in quantum physics problems. In some instances, you may want to shift things so that the square well is symmetric around the origin.
For example, the standard infinite square well looks like this:
But what if you want to move the square well so that it extends from
Here's what the new infinite square well looks like in this case:
You can translate from this new square well to the old one by adding a/2 to x, which means that you can write the wave function for the new square well in this equation as follows:
Doing a little trig gives you the following equations:
So as you can see, the result is a mix of sines and cosines.
Quantum Physics
In quantum physics, you may need to simplify and split the Schrödinger equation for hydrogen. Here's the usual quantum mechanical Schrödinger equation for the hydrogen atom:
The problem is that you're taking into account the distance the proton is from the center of mass of the atom, so the math is messy. If you were to assume that the proton is stationary and that rp = 0, this equation would break down to the following, which is much easier to solve:
Unfortunately, that equation isn't exact because it ignores the movement of the proton, so you see the more-complete version of the equation in quantum mechanics texts.
Quantum Physics
There are plenty of free particles — particles outside any square well —in the universe, and quantum physics has something to say about them. The discussion starts with the Schrödinger equation:
Say you’re dealing with a free particle whose general potential, V(x) = 0. In that case, you’d have the following equation:
And you can rewrite this as
where the wave number, k, is
You can write the general solution to this Schrödinger equation as
If you add time-dependence to the equation, you get this time-dependent wave function:
That’s a solution to the Schrödinger equation, but it turns out to be unphysical.
Quantum Physics
In quantum physics, once you relate the angles of the scattered particles in the lab frame and the center-of-mass frame, you can translate the differential cross section — the bull's eye when you're aiming to scatter the particles at a particular angle — between the lab and center-of-mass frames.
The differential area
is infinitesimal in size, and it stays the same between the two frames.
Quantum Physics
Creation and annihilation may sound like big make-or-break-the-universe kinds of ideas, but they play a starring role in the quantum world when you're working with harmonic oscillators. You use the creation and annihilation operators to solve harmonic oscillator problems because doing so is a clever way of handling the tougher Hamiltonian equation.
In quantum physics, ket notation makes the math easier than it is in matrix form because you can take advantage of a few mathematical relationships. For example, here’s the so-called Schwarz inequality for state vectors:
This says that the square of the absolute value of the product of two state vectors,
is less than or equal to
This turns out to be the analog of the vector inequality:
So why is the Schwarz inequality so useful?
Quantum Physics
What do Dirac notation and the Hermitian conjugate have in common? They help physicists to describe really, really big vectors. In most quantum physics problems, the vectors can be infinitely large — for example, a moving particle can be in an infinite number of states. Handling large arrays of states isn’t easy using vector notation, so instead of explicitly writing out the whole vector each time, quantum physics usually uses the notation developed by physicist Paul Dirac — the Dirac or bra-ket notation.
Quantum Physics
In quantum physics, you need to know how to use linear operators. An operator A is said to be linear if it meets the following condition:
For instance, the expression
is actually a linear operator. In order to understand this, you need to know just a little more about what happens when you take the products of bras and kets.
Quantum Physics
In quantum physics, you can use operators to extend the capabilities of bras and kets. Although they have intimidating-sounding names like Hamiltonian, unity, gradient, linear momentum, and Laplacian, these operators are actually your friends.
Taking the product of a bra and a ket,
is fine as far as it goes, but operators take you to the next step, where you can extract physical quantities that you can measure.
Quantum Physics
Quantum physicists understand that matter exhibits wave-like properties, which means that matter, like waves, aren't localized in space. This fact inspired Werner Heisenberg, in 1927, to come up with his celebrated uncertainty principle.
You can completely describe objects in classical physics by their momentum and position, both of which you can measure exactly.
Quantum Physics
In quantum physics, when working with kets, it is useful to know how to use eigenvectors and eigenvalues. Applying an operator to a ket can result in a new ket:
To make things easier, you can work with eigenvectors and eigenvalues (eigen is German for “innate” or “natural”). For example,
is an eigenvector of the operator A if
The number a is a complex constant
Note what’s happening here: Applying A to one of its eigenvectors,
multiplied by that eigenvector’s eigenvalue, a.
Quantum Physics
Quantum physicists use large particle accelerators to discover more about particle scattering on the subatomic level. You can think of a scattering experiment in terms of particles in and particles out. The following figure shows an example where particles are being sent in a stream from the left and interacting with a target; most of them continue on unscattered, but some particles interact with the target and scatter.
In quantum physics, when working with a box potential, you can make things simpler by assuming that the box is actually a cube. In other words, L = Lx = Ly = Lz. When the box is a cube, the equation for the energy becomes
So, for example, the energy of the ground state, where nx = ny = nz = 1, is given by the following, where E111 is the ground state:
Note that there's some degeneracy in the energies; for example, note that
So E211 = E121 = E112, which means that the first excited state is threefold degenerate, matching the threefold equivalence in dimensions.
Quantum Physics
In quantum physics, you can solve for the allowable energy states of a particle, whether it is bound, or trapped, in a potential well or is unbound, having the energy to escape.
Take a look at the potential in the following figure. The dip, or well, in the potential, means that particles can be trapped in it if they don’t have too much energy.
In quantum physics, when you work with spin eigenstates and operators for particles of spin 1/2 in terms of matrices, you may see the operators Sx, Sy, and Sz written in terms of Pauli matrices,
Given that the eigenvalues of the S2 operator are
and the eigenvalues of the Sz operator are
you can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the z axis.
This Cheat Sheet is intended to supplement Quantum Physics For Dummies, 3rd edition, by Andrew Zimmerman Jones. It begins by reviewing some useful operators used in quantum mechanics calculations. Then it covers a useful method for solving the Schrödinger equation for the quantum wave function, and then how you can use that wave function to calculate probabilities in quantum physics.
Quantum Physics
If your quantum physics instructor asks you to solve for the wave function of the center of mass of the electron/proton system in a hydrogen atom, you can do so using a modified Schrödinger equation:
What you will find is that you can actually ignore
and go straight on to
Here’s how it works.
Because the Schrödinger equation contains terms involving either R or r but not both, the form of this equation indicates that it’s a separable differential equation.
Quantum Physics
Your quantum physics instructor may ask you to solve for the wave function for a made-up particle of mass m in a hydrogen atom. To do this, you can begin by using a modified Schrödinger equation that solves for large and small r:
Because the Schrödinger equation contains terms involving either R or r but not both, the form of this equation indicates that it’s a separable differential equation.
In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down.
The eigenvalues of the S2 operator are
and the eigenvalues of the Sz operator are
You can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the z axis.
Quantum Physics
In quantum physics, you can break the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations to make it easier to solve 3D problems. In one dimension, the time-dependent Schrödinger equation (which lets you find a wave function) looks like this:
And you can generalize that into three dimensions like this:
Using the Laplacian operator, you can recast this into a more compact form.
Quantum Physics
When you swap the order of operations for two particles in a multi-particle atom, this affects their wave function. Whether the wave function is symmetric under such operations gives you insight into whether the two particles can occupy the same quantum state.
Take a look at the general wave function for N particles:
In this case, you can think of symmetry in terms of the location coordinate, r, to keep things simple, but you can also consider other quantities, such as spin, velocity, and so on.
Quantum Physics
In quantum physics, you sometimes need to use spherical coordinates instead of rectangular coordinates. For example, say you have a 3D box potential, and suppose that the potential well that the particle is trapped in looks like this, which is suited to working with rectangular coordinates:
Because you can easily break this potential down in the x, y, and z directions, you can break the wave function down that way, too, as you see here:
Solving for the wave function gives you the following normalized result in rectangular coordinates:
The energy levels also break down into separate contributions from all three rectangular axes:
E = Ex + Ey + Ez
And solving for E gives you this equation:
But what if the potential well a particle is trapped in has spherical symmetry, not rectangular?
Quantum Physics
In quantum physics, when you are working in one dimension, the general particle harmonic oscillator looks like the figure shown here, where the particle is under the influence of a restoring force — in this example, illustrated as a spring.
A harmonic oscillator.
The restoring force has the form Fx = –kxx in one dimension, where kx is the constant of proportionality between the force on the particle and the location of the particle.
Quantum Physics
This article takes a look at a 3D potential that forms a box, as you see in the following figure. You want to get the wave functions and the energy levels here.
A box potential in 3D.
Inside the box, say that V(x, y, z) = 0, and outside the box, say that
So you have the following:
Dividing V(x, y, z) into Vx(x), Vy(y), and Vz(z) gives you
Okay, because the potential goes to infinity at the walls of the box, the wave function,
must go to zero at the walls, so that's your constraint.
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