The Heisenberg Picture

Previously, we have used the Schrödinger picture to describe the dynamics of a system. This involves solving the time-dependent Schrödinger equation to find the time-evolution operator , which describes how the state of the system evolves over time. In this section, we will introduce the Heisenberg picture, which is an alternative formulation of quantum mechanics.

Table of Contents

• Table of Contents
• Introduction
• Heisenberg Equation of Motion
• Time-Evolution of Base Kets
• An Explicit Form of the Hamiltonian (Ehrenfest's Theorem)
• Summary and Next Steps

Introduction

The key difference between the two pictures is how we treat the operators and the states. This can be illustrated by considering an unitary operator and two states and . When we take the inner product of the two states along with some operator , we have:

This means that the inner product is invariant under the time-evolution operator, which is a direct result from the fact that is unitary. Now the key consideration is how we interpret this inner product; there are two approaches:

(These are equivalent from the associative axiom.)

• The left-hand side takes the approach of remaining constant while .
• The right-hand side takes the approach of while remains constant.

The first approach is the Schrödinger picture, where the operators are time-independent and the states evolve in time. The second approach is the Heisenberg picture, where the operators evolve in time and the states are time-independent. This is a matter of convention, and both pictures are equivalent. To take an example, consider taking the expectation value of at a certain time , :

• In the Schrödinger picture, we have:

• In the Heisenberg picture, we have:

The Heisenberg picture is more closely related to classical mechanics because it treats the operators as observables that evolve in time, similar to how classical quantities evolve in time. To distinguish between the two, we use the superscript to denote the Heisenberg picture and the superscript to denote the Schrödinger picture:

The following table summarizes the differences between the two pictures:

	Schrödinger Picture	Heisenberg Picture
Operators	Time-independent	Time-dependent
States	Time-dependent	Time-independent
Expectation Value

Heisenberg Equation of Motion

Now one may wonder what the Heisenberg-picture equivalent of the Schrödinger equation is. First, we rearrange Schrödinger's equation to isolate the derivative of :

To find the time derivative of , we can take the Hermitian conjugate of the above equation to get:

Now, we can take the time derivative of . Keep in mind we are assuming that is time-independent (which is true for most cases):

Since , we can add to both terms to be able to substitute :

Notice that outside of the common factors, the only difference between the two terms is the order of the operators. As such, we can rewrite the equation with the commutator :

Recall that in the Hamiltonian picture, operators evolve over time as . However, since in simple cases , we can see that commutes with . Thus and we can write:

This is the Heisenberg equation of motion, which describes how the operators evolve in time.

If one has studied Hamiltonian mechanics, this should look familiar. Hamilton's equations of motion are given by:

where denotes the Poisson bracket.

Time-Evolution of Base Kets

In the Schrödinger picture, state vectors evolve in time while their base kets and operators remain constant. On the other hand, in the Heisenberg picture, base kets do change in time. To begin with, suppose we have an operator with eigenket :

In the Schrödinger picture, is time-independent, and hence its eigenkets are naturally also time-independent. However, we have in the Heisenberg picture that . At time , the two pictures are equivalent, so we can use the eigenvalue equation above in both. Begin by applying the eigenvalue equation to :

Then, insert to the middle of the left-hand side;

We see the left-hand side is just , so we can write:

This implies that is an eigenket of with eigenvalue . As such, we deduce that base kets evolve in time as:

In other words, because of the on the time-evolution operator, these base kets rotate opposite to the state vectors.

The coefficient of the state vector in the Schrödinger picture is given by:

whereas in the Heisenberg picture, the coefficient is given by:

because is the dual of . We see that both are equivalent.

An Explicit Form of the Hamiltonian (Ehrenfest's Theorem)

Now that we have both the Schrödinger and Heisenberg equations of motion, we can start applying them to real systems. Notice that both of them rely on the Hamiltonian , but we do not currently have an expression for it. We use a few different ways to derive this expression:

1. Typically, we assume that is just the quantum equivalent of the classical Hamiltonian, but with the classical variables replaced with operators.
2. If, using the above method, there are noncommuting observables, we resolve this ambiguity by requiring that the Hamiltonian be Hermitian.
3. If there is no classical equivalent for the physical system, we make an educated guess for the Hamiltonian based on properties of the system. Then, we use experiments to determine its validity.

We will study a particle in one dimension, which is a simple system that has a well-defined classical Hamiltonian. This acts as a sanity-check for quantum mechanics; we will derive results that closely match the equations for classical mechanics:

• Momentum is conserved for a free particle.
• For a free particle, position evolves in time as .
• Newton's second law: .

First, we will derive two important results regarding the commutator of or with functions of and :

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Proof. We assume that can be expressed as a Taylor series expansion in terms of :

(If this appears unfamiliar, this is the Taylor series for a function of several variables.) To simplify the notation, we can use a multi-index notation, where and . Then we can write:

Our strategy is to first prove the result for a single monomial term , and then generalize it to the full Taylor series.

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The Hamiltonian for a free particle is the same as the classical Hamiltonian:

Taking any momentum operator and applying the Heisenberg equation of motion, we have:

The commutator is zero because is a function of . Thus, we have that momentum is conserved for a free particle, satisfying our first classical sanity-check.

If we take the position operator and apply the Heisenberg equation of motion, we have:

Now we use the result that :

where we have used the fact that . This means that , which is the classical equation of motion for a free particle;

This satisfies our second classical sanity-check.

Lastly, suppose we now place the particle in a potential . We can write the Hamiltonian as:

We will use the result that to find the equation of motion for . Applying Heisenberg's equation of motion, we have:

For , we still have (the new term commutes with position and hence does not contribute). Finally, we apply Heisenberg's equation of motion to :

But because this is just Heisenberg's equation applied to , we have:

We know that , so we can write:

Finally, if we apply this to all directions, we obtain a vector form of the equation:

This is the quantum equivalent of Newton's second law, which states that the acceleration of a particle is equal to the force acting on it divided by its mass. This satisfies our third classical sanity-check. However, this equation still explicitly uses operators, and hence only applies to the Heisenberg picture. To generalize this, we can multiply both sides by and take the expectation value of both sides:

Because expectation values are the same in both pictures, this equation is more general. It is known as Ehrenfest's theorem.

Summary and Next Steps

In this section, we have introduced the Heisenberg picture of quantum mechanics, which is an alternative formulation to the Schrödinger picture. The differences between the pictures are summarized in the table below.

	Schrödinger Picture	Heisenberg Picture
Operators	Time-independent	Time-dependent
States	Time-dependent	Time-independent
Base Kets	Time-independent	Time-dependent
Equation of Motion

We also derived Ehrenfest's theorem, an important sanity-check for quantum mechanics:

In the next section, we will study Feynman's path integral formulation of quantum mechanics.