4  From Classical Quantities to Quantum Operators

We laid the theoretical framework underpinning quantum mechanics in the formalism and applied to the real world with the postulates. At long last, we are able to start developing some physics.

Why all this setup? Many undergraduate courses in quantum mechanics will skip over all of this formal development at the beginning in favor of jumping straight into working with the wavefunction and utilizing the Schrödinger equation to solve basic problems. This choice is typically made to avoid adding needless complexity and to prevent the technical mathematical details from getting in the way of a basic physical intuition. As a matter of fact, this is precisely how we learned quantum mechanics initially, before supplementing the finer mathematical formalism in later graduate courses.

We considered following suit and also omitting these details to jump straight into the physics, but decided to include this for a few reasons. Primarily, we did this because this is not a university course on quantum mechanics. We are not bound by any time constraints and do not need to adhere to any explicit schedule, so the reader can take as much time as is required for everything to make sense. Since we don’t need to move through a unit within a week or two, there’s no longer a pressing need to omit certain details in the name of expediting the material.

Remark. This is not in any way suggesting that the way undergraduate courses structure quantum mechanics is wrong. Rather, since we are not bound by the same restrictions that an undergraduate course is, we can take certain liberties in material presentation.

Since we don’t have the same time restrictions, we can build up the material a bit more rigorously and mathematically. Even though this rigorous development requires a certain level of mathematical maturity and is ultimately more challenging for a first reader than jumping straight into the physics would be, we feel that it is extremely satisfying to see all of the operators and equations we use pop out directly from first principles and experimental results. This way, we don’t need to simply state the form of an operator without justification simply because we want to practice using it. Thus, we hope you have beared with us while we laid all the pieces in place.

As we warned at the beginning of the Formalism chapter, there is a lot of mathematics involved in the development of a formal framework for a physical theory. If the reader wishes to gain a more solid mathematical footing before attempting to understand certain parts of the way we present the material, that’s totally fine! We will wait here for you to return. It is our hope, though, that we do a sufficiently good job of presenting whatever mathematics is necessary throughout the relevant chapters and accompanying appendices at the back.

4.1 Position Representation

Let’s start with figuring out how to describe a particle’s position in quantum mechanics. For simplicity, we will start with the 1-dimensional case, then address 3 dimensions later on. Let’s also assume that a particle’s position in space (call it \(x\)) can be measured, and these measurement results are continuous (since space isn’t discretized). Let’s then denote the operator associated with measuring the position \(\hat{x}\) and any eigenkets of the position \(\ket{x}\). Thus, our eigenvalue problem takes the following form: \[\hat{x}\ket{x} = x\ket{x}\]

We will see much more of this hat notation moving forward; an operator will typically be denoted with a hat on top of it to indicate that it is an operator rather than a variable. The main exceptions will be those operators labeled with capital letters, since those are harder to confuse for variables or other mathematical objects.

Since we are dealing with a continuous spectrum and nondegenerate eigenkets (positional values can only correspond to a unique position), the resolution of the identity for this operator takes the following form: \[\int_{-\infty}^{\infty}\ketbra{x}dx = 1\] Now, let \(\ket{\psi}\) represent a normalized pure state for our system, and \(I=[x_0,x_1]\) be some interval on the \(x\)-axis. In this case, the projector onto the interval looks like this: \[P_I = \int_{x_0}^{x_1}\ketbra{x}dx\] Then, the probability postulate tells us that the probability of measuring some positional value within this interval is given by the following expectation value: \[\mathrm{Prob}(x_0\leq x\leq x_1) = \expval{P_I}{\psi} = \int_{x_0}^{x_1}\braket{\psi}{x}\braket{x}{\psi}dx\] Now, that last expression looks like we’re taking some squared norm. To make our lives easier (and to make this expression look more like an integral expression wherein we’re actually integrating a function), let’s finally define the positional wavefunction \(\psi(x)\):

Definition 4.1 (Positional Wavefunction) The wavefunction, denoted \(\psi(x)\), encodes the state in configuration space and is given by the following inner product: \[\psi(x) = \braket{x}{\psi}\]

Then, our probability becomes: \[\mathrm{Prob}(x_0\leq x \leq x_1) = \int_{x_0}^{x_1}|\psi(x)|^2dx = \int_{x_0}^{x_1}\psi^*(x)\psi(x)dx\] Here, we see that the squared norm of the positional wavefunction encodes the probability density of finding the particle at a given point in space. Now, we finally have a way to go from kets to functions!

Can we go the other way? No problem! If we multiply both sides of the resolution of the identity by \(\ket{\psi}\), we get a representation of the ket: \[\ket{\psi} = \int \ket{x}\braket{x}{\psi}dx \implies \boxed{\ket{\psi} = \int \ket{x}\psi(x)dx}\]

Now that we have a way to convert between ket formalism and functional notation, we also have a way to describe the inner product between two kets \(\ket{\psi}\) and \(\ket{\phi}\) by once again inserting the resolution of the identity between them: \[\braket{\psi}{\phi} = \int \braket{\psi}{x}\braket{x}{\phi}dx = \int \psi^*(x)\phi(x)dx\]

4.1.1 Generalization to 3 Dimensions

This formalism extends quite easily to three dimensions, where we simply have to consider three separate positional components \((x,y,z)\).

For the sake of consistency (and because it will allow us to make more efficient use of index notation), instead of labeling the components \((x,y,z)\), we will label them \((x_1,x_2,x_3)\). This way, we can compare arbitrary spatial components \(x_i\), \(x_j\) without having to find some random combination of \(x\), \(y\), and \(z\).

In this case, rather than simply measuring the overall position, we will have separate operators for measuring each component of the position: \(\hat{\mathbf{x}} = (\hat{x}_1, \hat{x}_2, \hat{x}_3)\). This is what’s known as a “vector operator”, since it’s a vector comprised of different operators. We will assume that the components all commute with each other (since they’re orthogonal and shouldn’t affect each other’s measurements): \[[\hat{x}_i,\hat{x}_j] = 0\] So, we now can write three separate eigenvalue problems, conveniently condensed into one using index notation: \[x_i\ket{\mathbf{x}} = x_i\ket{\mathbf{x}}\] Just as with the one-dimensional case, we have a similar resolution of the identity: \[1 = \int \ketbra{\mathbf{x}}d^3\mathbf{x}\] Then, our wavefunction-ket transformations translate seamlessly from the one-dimensional case: \[\psi(\mathbf{x}) = \braket{\mathbf{x}}{\psi}, \quad \ket{\psi} = \int \ket{\mathbf{x}}\psi(\mathbf{x})d^3\mathbf{x}\] Finally, the probability density of finding the particle in some region \(R\) in 3-dimensional space becomes a multivariate integral: \[\mathrm{Prob}(\mathbf{x}\in R) = \int_R|\psi(\mathbf{x})|^2d^3\mathbf{x}\]

4.1.2 The Position Operator

What we’ve accomplished thus far amounts to representing an arbitrary ket state in configuration space; this, at its core, is what the wavefunction is. What happens now when we try to represent the wavefunction of an operator’s action on a ket? For instance, if \(\ket{\psi}\) corresponds to \(\psi(x)\), what wavefunction does \(\hat{x}\ket{\psi}\) correspond to? Let’s label the resultant ket \(\ket{\phi} = \hat{x}\ket{\psi}\) and find its wavefunction representation: \[\phi(x) = \braket{x}{\phi} = \braket{x}(\hat{x}\ket{\psi}) = \mel{x}{\hat{x}}{\psi} = (x\bra{x})\ket{\psi} = x\braket{x}{\psi} = x\psi(x)\] Interestingly, acting the \(\hat{x}\) operator on a ket \(\ket{\psi}\) just multiplies the corresponding configuration space wavefunction by \(x\): \[\boxed{(\hat{x}\psi)(x) = x\psi(x)}\] Thus, in configuration space, \(\hat{x}\) is represented by multiplication of the wavefunction by \(x\). This is our position operator. Generalizing to three dimensions is easy: \[(\hat{x}_i\psi)(\mathbf{x}) = x_i\psi(\mathbf{x})\]

Remark. There’s one lingering question here: what’s the physical meaning of the position eigenkets \(\ket{x}\), \(\ket{\mathbf{x}}\)? As we discovered before, these are not normalizable, which means they cannot belong to any relevant Hilbert space that we could make use of. This isn’t actually a big deal for us, because we’ll never measure the position of a particle exactly; rather, we will localize it to some region of space, and the eigenkets of position are simply an idealized variant where the size of the region is sent to \(0\) in the limit.

In general, measuring the position of a particle is an inherently nonrelativistic concept, since localizing the position to smaller and smaller regions will blow up the momentum (because of the uncertainty relation which we’ll quantify more explicitly later), until it eventually reaches relativistic ranges. As such, the concept of a position operator and measuring a particle’s position will be relegated mostly to nonrelativistic quantum mechanics (which we’ll be developing in the first two parts of this book), with relativistic quantum mechanics (explored in Part III) requiring a fundamentally different treatment.

4.2 Momentum Representation

Now that we’ve established how to appropriately make positional measurements, a natural next step is to understand how to quantify momentum and associated momentum measurements.

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4.2.1 Translation Operators

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4.2.2 Generators of Translations

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4.2.3 Classical vs. Quantum Momentum

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4.2.3.1 Classical Momentum

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4.2.3.2 Quantum Momentum

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4.2.4 Canonical Commutation Relation

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4.3 The Schrödinger Equation

We are now finally ready to develop the Schrödinger equation, which will serve as the main equation all of quantum mechanics is aimed to solve to describe physical systems. To do so properly, however, we need to quantify how to determine a system’s energy in a quantum-mechanical system, which will require us to find the quantum analog of the Hamiltonian. We set everything up in a similar manner to how we presented (and ultimately derived) the momentum.

4.3.1 Time-Evolution Operator

At their core, any general equation for a specified field of physics describes how that particular system evolves in time. Some systems will not change in time, but being able to quantify how certain properties change as time advances is the core of that particular field. In classical mechanics, we saw that first through Newton’s Laws of Motion, which described how forces acting on an object in a system affect said object. Later on, we quantified how to describe the change in a system’s energy, for which we used the Lagrangian/Hamiltonian Equations of Motion. In quantum mechanics, we want to something very similar; we wish to understand how to express how a system evolves in time by quantifying its energy.

Let’s now finally introduce time dependence into our ket vectors. Label the pure state of a system at some initial time \(t_0\) by \(\ket{\psi(t_0)}\), and label that same state at a final time \(t\) by \(\ket{\psi(t)}\). We now claim the existence of a time-evolution operator, which we’ll denote \(U(t,t_0)\), that transforms the state \(\ket{\psi(t_0)}\) into the state \(\ket{\psi(t)}\): \[\ket{\psi(t)} = U(t,t_0)\ket{\psi(t_0)} \tag{4.1}\] Let’s properly define this operator:

Definition 4.2 (Time-Evolution Operator) The time-evolution operator, denoted \(U(t,t_0)\), evolves a quantum state from initial time \(t_0\) to final time \(t\), while satisfying the following three properties:

1. \(U\) should reduce to the identity operator at \(t=t_0\): \[U(t_0,t_0) = 1\]
2. \(U\) is unitary: \[U(t,t_0)^{-1} = U(t,t_0)^{\dagger}\]
3. \(U\) should satisfy the composition property: \[U(t_2,t_1)U(t_1,t_0) = U(t_2,t_0)\]

Each of these properties make intuitive sense. Starting with the first, evolving between the same time is equivalent to not evolving in time at all, meaning the states before and after such a time evolution action should be equivalent. Secondly, we want to preserve probabilities. In other words, the probability of something like finding a particle somewhere in space should be \(1\) regardless of when we measure in time. Thus, evolving forwards vs. backwards in time shouldn’t affect such probabilities. Finally, evolving a state from an initial time to a second time and then evolving to a final third should naturally produce the same result as directly evolving from the initial time to the final.

4.3.2 The Hamiltonian

Let’s now consider some infinitesimal time evolution from \(t\) to \(t+\epsilon\), for some small \(\epsilon\). We can expand \(U\) to first order and tack on an additional correction term: \[U(t+\epsilon,t) = 1 - i\epsilon\Omega(t) + \dots \tag{4.2}\] Our correction term is defined as follows (with an extra \(i\) factor to make the operator Hermitian: \[\Omega(t) = i\pdv{t'}U(t',t)\bigg|_{t'=t} \tag{4.3}\] This operator \(\Omega(t)\) is then our generator of time evolution in quantum mechanics, since it is precisely the correction term that advances a state in time by small increments. Combining these terms in large quantities will advance time by larger increments.

In classical mechanics¹, the Hamiltonian (which we’ll henceforth represent with \(H\)) is fundamentally the generator of time translations, since it describes how a system evolves in time. Let’s consider some classical observable \(f(x,p)\) (a function in position and momentum) that exhibits time dependence due to how \(x\) and \(p\) evolve in time. Expanding to first order in \(t\) gives: \[\begin{align*} f(t+\epsilon) &= f(t) + \epsilon\dv{f}{t} = f(t)+\epsilon\left(\dot{x}\pdv{f}{x} + \dot{p}\pdv{f}{p}\right) = \\ &= f(t) + \epsilon\left(\pdv{H}{p}\pdv{f}{x} - \pdv{H}{x}\pdv{f}{p}\right) = f(t) - \epsilon\{H,f\} \end{align*}\] Here, \(\{H,f\}\) gives the Poisson bracket (discussed in more detail in Section 4.6.1).

Since we wish to develop a quantum analog to the classical Hamiltonian, it would make sense for the quantum Hamiltonian to also function as a generator of time evolution, meaning we want the associated Hamiltonian operator (henceforth denoted \(\hat{H}\)) to be related to the generator of time evolutions defined in Equation 4.3: \[\hat{H}(t) = \hbar\Omega(t)\] We can plug this back into Equation 4.2 to express the time evolution operator exclusively in terms of the Hamiltonian: \[U(t+\epsilon,t) = 1 - \frac{i\epsilon}{\hbar}\hat{H}(t) + \dots\]

Remark. Classical Hamiltonians will depend on time in the presence of time-dependent forces or fields. As such, we will generally expect the quantum Hamiltonian \(\hat{H}\) to also be time-dependent. There will still, however, be special time-independent cases that we will choose to study closely (this may or may not be foreshadowing).

It is, of course, a nontrivial question whether or not this \(\hbar\) constant is the same one that is used in the de Broglie relations and the subsequent derivation of the quantum momentum done in Section 4.2. There is a requirement that these \(\hbar\)’s be equivalent (known as relativistic covariance), but this won’t be relevant to us until we discuss relativistic quantum mechanics. To avoid a lengthy and complicated discussion, we will simply note here that this \(\hbar\) constant is the same as the one that appears in momentum and energy representations.

4.3.3 Deriving the Schrödinger Equation

With that established, we now need a differential equation that we can solve to ascertain physical properties of a system. Classically, that would involve Hamilton’s equations of motion (or Newton’s or Lagrange’s, depending on the scenario). So, let’s find a differential equation for time-evolution operators. Let’s start with the basic definition of the derivative: \[\pdv{U(t,t_0)}{t} = \lim_{\epsilon\to0}\frac{U(t+\epsilon,t_0) - U(t,t_0)}{\epsilon}\] The composition property in Definition 4.2 allows us to rewrite \(U(t+\epsilon,t_0)\): \[U(t+\epsilon,t_0) = U(t+\epsilon,t)U(t,t_0)\] Substituting this back into our definition of the derivative and factoring out \(U(t,t_0)\) gives: \[\pdv{U(t,t_0)}{t} = \left(\lim_{\epsilon\to0}\frac{U(t+\epsilon,t) - 1}{\epsilon}\right)U(t,t_0) = \pdv{U(t',t)}{t'}\bigg|_{t'=t}U(t,t_0)\] By Equation 4.3, the RHS derivative can be replaced with \(-i\Omega(t)\), which in turn can be replaced with \(\frac{-i\hat{H}(t)}{\hbar}\). Assembling all of that together (and rearranging a few prefactors) gives us our differential equation for the time evolution operator in terms of the Hamiltonian: \[i\hbar\pdv{U(t,t_0)}{t} = \hat{H}(t)U(t,t_0) \tag{4.4}\] Now, we can finally get rid of \(U\) by differentiating Equation 4.1 with respect to time and substituting in this new expression to get rid of the time evolution (remember also to tack on the relevant \(i\hbar\) prefactor): \[\begin{align*} i\hbar\pdv{t}\ket{\psi(t)} &= i\hbar\pdv{t}\left(U(t,t_0)\ket{\psi(t_0)}\right) = i\hbar\left[\pdv{U(t,t_0)}{t}\ket{\psi(t_0)} + \cancelto{0}{U(t,t_0)\pdv{t}\ket{\psi(t_0)}}\right] = \\ &= i\hbar\pdv{U(t,t_0)}{t}\ket{\psi(t_0)} = \hat{H}(t)U(t,t_0)\ket{\psi(t_0)} = \hat{H}(t)\ket{\psi(t)} \end{align*}\] We summarize this as the Schrödinger equation:

Theorem 4.1 (Schrödinger Equation) \[i\hbar\pdv{t}\ket{\psi(t)} = \hat{H}(t)\ket{\psi(t)}\]

More precisely, this is the time-dependent Schrödinger equation, since we are allowing for a time-dependent Hamiltonian. This is the most general form of the equation, since it makes no claims as to what form \(\hat{H}\) takes and it doesn’t express \(\ket{\psi}\) in any particular basis. Can we be a bit more precise about the Hamiltonian? As a matter of fact, we can!

Classically, the Hamiltonian of a system is given as the sum of its kinetic and potential energies² (ie, the total energy of the system): \[H = T + V\] Here, \(T\) represents the kinetic energy, while \(V\) represents the potential energy. The kinetic energy is given by \(\frac{p^2}{2m}\), and the potential energy is generally unknown and will be system-depedent. Translating all respective quantities to quantum operators gives us the quantum Hamiltonian: \[\hat{H} = \frac{\hat{p}^2}{2m} + \hat{V}\] Translating this into configuration space wavefunction notation gives us: \[\hat{H} = -\frac{\hbar^2}{2m}\pdv[2]{x} + V(x,t)\] Generally, the potential will be time-dependent. Substituting all of that in gives us a slightly more specific form of the Schrödinger equation that we will actually apply to solve problems:

Theorem 4.2 (Time-Dependent Schrödinger Equation) \[\left(\frac{\hbar^2}{2m}\pdv[2]{x} + V(x,t)\right)\psi(x,t) = i\hbar\pdv{\psi(x,t)}{t}\]

To be a bit more precise, this was the one-dimensional form of the Schrödinger equation. In three dimensions, we replace the derivatives with Laplacians: \[\left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{x},t)\right)\psi(\mathbf{x},t) = i\hbar\pdv{\psi(\mathbf{x},t)}{t}\] This is only one representation of the Schrödinger equation in functional form. There are, of course, many other bases in which we can represent (and subsequently solve) this equation. The most natural alternat form is writing the Schrödinger equation in momentum space, which we can do with a simple Fourier transform.

4.3.4 Time-Independent Schrödinger Equation

While wavefunctions, potentials, and Hamiltonians will generally admit time dependence, it is also worth analyzing some time-independent potentials as a special case and a useful model for certain problems in nonrelativistic quantum mechanics. As such, we can write a time-independent version of the Schrödinger equation, where we instead focus on finding the energy eigenstates (or stationary states, since they don’t evolve in time). Thus, we can represent the Schrödinger equation as an eigenvalue problem:

Theorem 4.3 (Time-Independent Schrödinger Equation) \[\hat{H}\ket{\psi} = E\ket{\psi}\]

Alternatively, you can enumerate the eigenstates \(\ket{n}\) and the associated energy eigenvalues \(E_n\): \[\hat{H}\ket{n} = E_n\ket{n}\] Crucially, \(n\) can represent one or more quantum numbers required to specify the energy eigenstate.

Remark. This will become much more relevant in higher dimensions, where we will have radial and angular components, etc.

For the time being, much of our work will involve solving the time-independent Schrödinger equation for various relatively simple time-independent potentials. Should we choose to reintroduce time dependence, we will be able to do so with a separate expression that stores all of the information on the system’s time dependence (this is explored in more detail in the next chapter).

4.4 Pictures in Quantum Mechanics

As it turns out, there are multiple different interpretations for how operators and wavefunctions behave. So far, we have been operating in only one of these so-called pictures, but it is useful to quantify the difference between the two of them and see how each may be more applicable depending on the type of problem we are trying to solve.

4.4.1 Schrödinger Picture

So far, we have been operating under the assumption that the wavefunctions are what evolve in time, with observables remaining fixed. In other words, the way we measure position will never depend on what point in time we choose to measure that position. This is what’s known as the Schrödinger picture.

Since our wavefunction is what describes the state of the system, it is natural for it to be the quantity that evolves in time. As a consequence, most problems we will tackle and most attempts at solving the Schrödinger equation will be undertaken in the Schrödinger picture.

In this section, we denote operators and kets in the Schrödinger picture with a subscripted \(S\): \[\ket{\psi} \to \ket{\psi_S(t)}, \quad \hat{A} \to \hat{A}_S\] In general, if a subscript is not present, then we assume that we are working in the Schrödinger picture.

4.4.2 Heisenberg Picture

That being said, there is another way we can interpret systems. What if, instead of letting the wavefunction evolve in time whilst keeping observables static, we go the other way: let observables evolve in time whilst keeping wavefunctions static. This is what’s known as the Heisenberg picture, with our equations of motion now involving time-derivatives of operators rather than wavefunctions.

We denote operators and kets in the Heisenberg picture with a subscripted \(H\): \[\ket{\psi} \to \ket{\psi_H}, \quad \hat{A} \to \hat{A}_H(t)\]

Since we are eliminating time dependence from kets in the Heisenberg picture, we relate them to the corresponding time-dependent state vector in the Schrödinger picture by a time evolution: \[\ket{\psi_H} = U(t,t_0)^{\dagger}\ket{\psi_S(t)}\] This is equivalent to writing that the Heisenberg state vector is simply the Schrödinger state vector at the initial time \(t_0\): \[\boxed{\ket{\psi_H} = \ket{\psi_S(t_0)}} \tag{4.5}\] Now, operators in the Schrödinger picture are still generally allowed to have an explicit time dependence, we need to introduce a separate (implicit) time depedence when relating to the Heisenberg operator: \[\hat{A}_H(t) = U(t,t_0)^{\dagger}\hat{A}_S(t)U(t,t_0) \tag{4.6}\] Since the Heisenberg kets don’t have any time dependence, we no longer have a time evolution equation for them like we did in the Schrödinger picture. Instead, we need a time evolution equation for Heisenberg operators. If we differentiate the above equation and once again invoke Equation 4.4, we get: \[i\hbar\dv{\hat{A}_H}{t} = -U^{\dagger}(t,t_0)\hat{A}_SU(t,t_0) + U^{\dagger}(t,t_0)\hat{A}_S\hat{H}U(t,t_0) + i\hbar U^{\dagger}(t,t_0)\pdv{\hat{A}_S}{t}U(t,t_0)\] Notice, however, that the first two terms on the RHS can be converted to their Heisenberg equivalents by Equation 4.6: \[-U^{\dagger}(t,t_0)\hat{H}\hat{A}_SU(t,t_0) + U^{\dagger}(t,t_0)\hat{A}_S\hat{H}U(t,t_0) = -\hat{H}_H\hat{A}_H + \hat{A}_H\hat{H}_H = [\hat{A}_H,\hat{H}_H]\] The final term on the RHS can be rewritten in a similar manner to represent the derivative of the operator \(\hat{A}\) in the Heisenberg picture: \[i\hbar U^{\dagger}(t,t_0)\pdv{\hat{A}_S}{t}U(t,t_0) = i\hbar\left(\pdv{\hat{A}}{t}\right)_H\] Putting these together gives us the Heisenberg equations of motion:

Theorem 4.4 (Heisenberg Equations of Motion) \[i\hbar\dv{\hat{A}_H}{t} = [\hat{A}_H,\hat{H}_H] + i\hbar\left(\pdv{\hat{A}}{t}\right)_H\]

If the prefactors are annoying, we can instead write it as: \[\dv{\hat{A}_H}{t} = -\tfrac{i}{\hbar}[\hat{A}_H,\hat{H}_H] + \left(\pdv{\hat{A}}{t}\right)_H\] Whenever we operate in the Heisenberg picture, Theorem 4.4 will be the equation(s) we will be interested in solving rather than anything to do with kets.

4.4.3 Equivalence of the Pictures

Despite the fact that these two pictures seem fundamentally different and involve letting two completely different types of objects hold time dependence, they actually turn out to be equivalent. This makes sense, because the solution to a particular problem shouldn’t depend on the method used to solve that problem. Instead of merely saying that, however, let’s take the time now to prove that algebraic relations are equivalent between the two pictures.

In the previous section, we saw how to convert kets and operators between the Schrödinger and Heisenberg pictures, but what about algebraic relations between operators/kets? Let’s consider some Schrödinger operator \(\hat{A}_S\), related to the Heisenberg operator \(\hat{A}_H\) by the usual \(\hat{A}_H = U^{\dagger}\hat{A}_SU\) (here, \(U=U(t,t_0)\), where we dropped the functional notation for brevity). Now suppose we have a different Schrödinger operator \(\hat{B}_S = f(\hat{A}_S)\), where \(f\) is some function. Let’s see what \(\hat{B}_H\) would look like: \[\hat{B}_H = U^{\dagger}f(\hat{A}_S)U = f(U^{\dagger}\hat{A}_SU) = f(\hat{A}_H)\] Wait, hold on a second; why can we just pull the \(U\)’s into the function? We can start by convincing ourselves that this true for simple forms of \(f\), such as a polynomial or power series. Take, for instance, \(f(\hat{A}_S) = \hat{A}^2_S\): \[\hat{B}_H = U^{\dagger}\hat{A}_S\hat{A}_SU = U^{\dagger}\hat{A}_SUU^{\dagger}\hat{A}_SU = \hat{A}_H\hat{A}_H = \hat{A}_H^2\] More generally, we can prove this result holds by recalling the definition of the function of an observable discussed in the Formalism chapter. We won’t go into too much detail here, so we conclude our discussion with that.

This relation equivalence also holds for products of operators that don’t necessarily commute. Suppose we had \(\hat{C}_S = \hat{A}_S\hat{B}_S\), where the latter two don’t necessarily commute. Then: \[\hat{C}_H = U^{\dagger}\hat{A}_S\hat{B}_SU = U^{\dagger}\hat{A}_SUU^{\dagger}\hat{B}_SU = \hat{A}_H\hat{B}_H\] Notably, we can combine the above two results to conclude that commutation relations also maintain the same form across both pictures: \[\hat{C}_S = [\hat{A}_S,\hat{B}_S] \implies \hat{C}_H = [\hat{A}_H,\hat{B}_H]\] Finally, potentials also have the same form across both pictures: \[U^{\dagger}V(\mathbf{x}_S)U = V(\mathbf{x}_H)\] Putting all of these together, we can conclude that the Hamiltonian itself has the same form in both pictures: \[\hat{H}_S = \frac{\mathbf{p}_S^2}{2m} + V(\mathbf{x}_S) \implies \hat{H}_H = \frac{\mathbf{p}_S^2}{2m} + V(\mathbf{x}_H)\] We have shown that algebraic relations are the same, but how do we show that the Heisenberg and Schrödinger pictures are physically equivalent? Well, we need to consider measurable quantities in quantum mechancis and see how they compare to each other across the two pictures. Each measurable quantity can be represented in terms of matrix elements of the form \(\mel{\phi}{\hat{A}}{\psi}\). Thus, if we can show that these are equivalent between the two pictures, we can conclude that either picture should predict (and subsequently yield) the same experimental outcomes. Let’s start in the Heisenberg picture: \[\begin{align*} \mel{\phi_H}{\hat{A}_H(t)}{\psi_H} &= (\bra{\phi_S(t_0)})(U^{\dagger}(t,t_0)\hat{A}_SU(t,t_0))(\ket{\psi_S(t)}) = \\ &= (\bra{\phi_S(t_0)}U^{\dagger}(t,t_0))\hat{A}_S(U(t,t_0)\ket{\psi_S(t)}) = \\ &= \mel{\phi_S(t)}{\hat{A}_S}{\psi_S(t)} \end{align*}\] Thus, matrix elements are equivalent between the two pictures, meaning any measurable quantity will be equivalent between these pictures as well. This means that we can predict the outcome of any experiment equally well by finding the time evolution of operators in the Heisenbeg picture of the time evolution of state vectors in the Schrödinger picture.

As is par for the course, it is worth considering a special case. In particular, when the Schrödinger Hamiltonian \(\hat{H}_S\) is time-independent, the time-evolution operator will be a function of \(H\) and thus commute with it. This then means that the Schrödinger and Heisenberg Hamiltonians will be equal: \[\hat{H}_H = U^{\dagger}\hat{H}_SU = U^{\dagger}U\hat{H}_S = \hat{H}_S\]

4.4.4 Other Pictures?

Are there other pictures beyond the two that we’ve considered just now? Yes! In particular, when we start dealing with something called time-dependent perturbation theory (in Part II), we will consider the interaction picture, which can be considered a sort-of intermediary between these two³.

4.5 Probability Current

[This section is currently incomplete; it will be updated very soon!]

4.6 Commutators

Lastly, even commutators can be derived from classical quantities! Previously, we introduced the notion of commutators as a way to quantify how close/far two operators are from commuting with each other. However, it is actually possible to connect this notion to classical mechanics.

4.6.1 Poisson Brackets

First, we will introduce the Poisson bracket, used extensively in classical mechanics (particularly with Hamiltonian mechanics):

Definition 4.3 (Poisson Bracket) The Poisson bracket of two functions \(f(p_i,q_i,t)\) and \(g(p_i,q_i,t)\), denoted \(\{f,g\}\), is given as: \[\{f,g\} = \sum_{i}\left(\pdv{f}{q_i}\pdv{g}{p_i} - \pdv{f}{p_i}\pdv{g}{q_i}\right)\]

Here, \(p_i\) represent components of the generalized momentum, \(q_i\) represent generalized coordinates, and \(t\) is time. Together, the make up canonical coordinates. We will not worry about what these mean exactly and save detailed discussion for a book on clasical mechanics.

The main thing we will note is that we can express the total time derivative of a classical observable \(A(p_i,q_i,t)\) as a sum of the explicit time dependence and implicit time dependence resulting from the time dependence of \(p\) and \(q\) using the Poisson bracket: \[\dv{A}{t} = \underbrace{\pdv{A}{t}}_{\text{explicit}} + \underbrace{\{A,H\}}_{\text{implicit}}\] This looks suspiciously similar to Theorem 4.4, and we can in fact create an explicit mapping between the Poisson bracket and the commutator to make this work: \[[A,B] \to i\hbar\{A,B\}\] We applaud Paul Dirac for noticing this correspondence.

There is one important caveat: this isn’t an exact correspondence in every scenario, since the commutator is still defined slightly differently. Fortunately for us, the correspondence will be exact in the most crucial scenarios. For instance, the Poisson bracket relations for the classical angular momentum \(\mathbf{L}=\mathbf{x}\times\mathbf{p}\) is given by: \[\{L_i,L_j\} = \epsilon_{ijk}L_k\]⁴ Likewise, as we will discover later, the quantum-mechanical commutation relations for the angular momentum become: \[[\hat{L}_i,\hat{L}_j] = i\hbar\epsilon_{ijk}\hat{L}_k\] In other cases, however, the correspondence will only be an approximation.

4.6.2 Ehrenfest’s Theorem

The last thing we’ll do in this chapter is compute the time derivative of the expectation value of an observable. Armed with the Heisenberg equations of motion and an understanding of how to take time derivatives of operators, let’s consider some arbitrary observable \(A\): \[\dv{t}\expval{A} = \pdv{t}\braket{\psi}{\hat{A}\psi} = \braket{\pdv{\psi}{t}}{\hat{A}\psi} + \braket{\psi}{\pdv{\hat{A}}{t}\psi} + \braket{\psi}{\hat{A}\pdv{\psi}{t}}\] Now, the Schrödinger equation tells us that we can replace \(i\hbar\pdv{t}\ket{\psi} = \hat{H}\ket{\psi}\). Substituting that in gives us: \[\dv{t}\expval{A} = -\tfrac{i}{\hbar}\braket{\hat{H}\psi}{\hat{A}\psi} + \tfrac{1}{i\hbar}\braket{\psi}{\hat{A}\hat{H}\psi} + \expval{\pdv{\hat{A}}{t}}\] Finally, since \(\hat{H}\) is Hermitian, we can move it through the inner product and we have that \(\braket{\hat{H}\psi}{\hat{A}\psi} = \braket{\psi}{\hat{H}\hat{A}\psi}\). Combining the first two terms on the RHS into a commutator, we summarize the final form of our equation as the generalized Ehrenfest theorem:

Theorem 4.5 (Generalized Ehrenfest’s Theorem) \[\dv{t}\expval{A} =\tfrac{i}{\hbar}\expval{[\hat{H},\hat{A}]} + \expval{\pdv{\hat{A}}{t}}\]

The term “generalized Ehrenfest theorem” to describe the above equation was coined in Griffiths and Schroeter (2018), as it doesn’t actually have an official name. Since, however, it is a useful expression that can be used to derive more specific relations that are named, we’ll use this term.

Let’s put this theorem to the test with two special cases, notably the position and momentum operators.

In this case, let’s work out the Heisenberg equations of motion for \(x\) and \(p\). Plugging them directly into Theorem 4.4 gives: \[\dv{t}\mathbf{x}_H = -\tfrac{i}{\hbar}[\hat{\mathbf{x}}_H,\hat{H}_H], \quad \dv{t}\mathbf{p}_H = -\tfrac{i}{\hbar}[\hat{\mathbf{p}}_H,\hat{H}_H]\] Since the position and momentum operators don’t have explicit time dependence, the final term drops out for both of them. Now, we can evaluate the commutators: \[\dv{t}\mathbf{x}_H = \frac{\mathbf{p}_H}{m}, \quad \dv{t}\mathbf{p}_H = -nabla V(\mathbf{x},t)\] Taking the expectation value with respect to some \(\ket{\psi}\) gives us two special cases of Theorem 4.5 which are collectively named Ehrenfest’s theorem:

Theorem 4.6 (Ehrenfest’s Theorem) \[\dv{t}\expval{\mathbf{x}} = \frac{\expval{\mathbf{p}}}{m}, \quad \dv{t}\expval{\mathbf{p}} = -\expval{\nabla V(\mathbf{x},t)}\]

If we so desire, we could eliminate \(\expval{\mathbf{p}}\) and express the latter relation purely in terms of \(\expval{\mathbf{x}}\): \[m\dv[2]{\expval{\mathbf{x}}}{t} = -\expval{\nabla V(\mathbf{x},t)}\]

4.7 Conclusion

As mentioned earlier, much of what appears in this chapter is generally omitted in a first course on quantum mechanics in favor of jumping straight into the physics and trying to describe systems with new equations. Perhaps this may have been easier, but at the end of the day, it is entirely about the journey rather than the destination. Since we are not beholden to any schedule and are attempting to organize these notes in a way that we felt works best, we have chosen to take the road less traveled. There is a light at the end of this tunnel, however; we are at last done with the difficult setup. The following chapters will take a much lighter approach that may be more akin to an undergraduate perspective, albeit with some formalism sprinkled in every now and then.

Let’s use our newly-acquired tools to finally start describing some basic one-dimensional systems!

Axler, Sheldon. 2023. Linear Algebra Done Right. Fourth Edition. Springer.

French, A. P., and E. F. Taylor. 2017. An Introduction to Quantum Physics (MIT Introductory Physics Series). First edition. CRC Press.

Griffiths, David J., and Darrell F. Schroeter. 2018. Introduction to Quantum Mechanics. Third edition. Cambridge University Press.

Littlejohn, Robert G. 2022. Physics 221AB (Quantum Mechanics) Lecture Notes. Https://bohr.physics.berkeley.edu/classes/221/2122/221.html.

Muller, Holger. 2023. Physics 221A/b Supplementary Lecture Notes.

Sakurai, Jun John. 2020. Modern Quantum Mechanics. Third edition. Cambridge University Press.

Schwabl, Franz, R. Hilton, and Angela Lahee. 2008. Advanced Quantum Mechanics. Fourth Edition. Springer.

Shankar, Ramamurti. 1994. Principles of Quantum Mechanics. Second Edition. Plenum Press.

Siddiqi, Irfan. 2022. Physics 137A Lecture Notes.

Taylor, John R. 2005. Classical Mechanics. University Science Books.

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1. This section in particular (among others that deal with the transition from classical to quantum mechanics) will make use of some topics in classical mechanics, but will not go in detail about their derivations or physical significance. Since this is not a text on classical mechanics, some prior knowledge on that matter is assumed for these sections. If these topics are unfamiliar, we encourage the reader to consult our notes on classical mechanics (when they’re written) or any text on classical mechanics (some of which are linked in our References).

2. Those who have studied classical mechanics will recognize that this isn’t exactly true generally: the Hamiltonian only describes the total energy of the system when we are working in a natural coordinate system (ie, the relation between the positional descriptions of particles in a system and the generalized coordinates is time-independent). In functionally all of our cases, though, we will simply choose natural coordinate systems, meaning we can safely take the Hamiltonian to represent the total energy. See (Taylor 2005) for a more in-depth description of generalized coordinates and how they affect the form of the Lagrangian and Hamiltonian.

3. Specifically, the time dependence due to some unperturbed base Hamiltonian \(\hat{H}_0\) will be moved from the kets to the operators, while the time dependence due to some perturbation \(\hat{H}_1\) will remain on the kets.

4. \(\epsilon_{ijk}\) is what’s known as the Levi-Civita tensor (or symbol). It is yet another tool used in index notation to compactify notation. We discuss this symbol (and how to effectively utilize it) in more detail in our appendices.