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Time Evolution of Quantum Systems

In this section, we will derive the third postulate of quantum mechanics, which requires that the time evolution of quantum systems be governed by a unitary operator.

Table of Contents​

Time-Evolution Operator​

The time-evolution operator is an operator that describes how a quantum system evolves in time. Suppose we have a quantum system described by a state vector at time . At some other time , the state of the system will be described by another state vector. Sakurai uses the notation to denote the state of the system at time given that it was in the state at time .

First, if , then the state of the system at time is the same as the state of the system at time . Mathematically, this is expressed as:

Next, suppose we have a time-evolution operator that describes how the state of the system evolves from time to time . Then, the state of the system at time is given by:

From physical intuition, we expect the time-evolution operator to follow the following properties:

  • Equation can be written as:

    where is the identity operator.

  • The time-evolution operator should be invertible because we should be able to evolve the state of the system backwards in time. This means that there should exist an operator such that:

  • The operator should follow this composition property:

  • The total probability of the system should be conserved. Since the total probability is equal to , we should have:

The last condition is very interesting. Recall that the Hermitian conjugate allows us to move to the right side of the inner product:

Since this must be equal to , we must have:

This property is very important and it has a name; the time-evolution operator must be an unitary operator. (Recall that we have already seen unitary operators in the context of the transformations of the Jones vectors in the polarization of light.)

Schrödinger Equation​

Consider a quantum system described by a state vector at time . Now suppose we want to find the state of the system at time (a very small time interval after ).

From Equation , we have:

Therefore, the time-evolution operator must be very close to the identity operator:

We can expand in a Taylor series about :

Therefore:

From , we have :

Finally, dividing by gives:

Now, we apply both sides of the equation to the state vector :

Taking the limit as gives:

This is the definition of the derivative of the state vector with respect to time. Thus:

To continue from here, consider the term . Since is a unitary operator, we have . Expanding this out in a Taylor series (about ) gives:

The terms cancel out, and we are left with (after dividing by and taking the limit as ):

Or,

This is the definition of an anti-Hermitian operator; we have previously seen it in the context of commutators.

The key insight comes when we try to evaluate . From the product rule for the Hermitian conjugate, we have:

In other words, , and so it must be Hermitian. Denote this operator as . Thus, we have:

So can be written as:

Going back to Equation , we have:

Rearranging to the left gives:

The Appendix outlines the connection between the time-evolution operator and the Hamiltonian of a system:

It looks very close to the previous equation given above; and . As such, we can define as the following:

Thus, we have:

Rearranging this gives the Schrödinger equation:

Schrödinger Equation: The time evolution of a quantum system is governed by the Schrödinger equation:

Alternatively, we can write the Schrödinger equation in terms of the time-evolution operator :

Schrödinger Equation (Time-Evolution Operator Form): The time evolution of a quantum system is governed by the Schrödinger equation:

The reason the in this context is equal to the in the translation operator is because it is required to create a relationship like:

Solving the Schrödinger Equation​

For the time-evolution operator form of the Schrödinger equation, we can solve it depending on the form of the Hamiltonian.

  1. Time-Independent Hamiltonian: If the Hamiltonian is time-independent, which corresponds to systems with fixed energies, then the time-evolution operator can be written as:

    When the Hamiltonian is time-independent, the Schrödinger equation is simply an exponential-like differential equation of the form .

  2. Time-Dependent, Commuting Hamiltonians: If the Hamiltonians at different times commute, then the time-evolution operator can be written as:

  3. Time-Dependent, Non-Commuting Hamiltonians: If the Hamiltonians at different times do not commute, then the time-evolution operator can be written as:

    This is known as the Dyson series, named after Freeman J. Dyson.

Time-Independent Hamiltonians​

For now on, we will consider the case where the Hamiltonian is time-independent. In this case, the Schrödinger equation simplifies to the first case:

To find out how acts on a state vector , it is useful to first know how it acts on the eigenstates of whatever operator we are considering. It is convenient to choose an operator such that . Then, the eigenstates of are also eigenstates of , known as energy eigenkets. Their eigenvalues are denoted by :

Then, applying the completeness relation to the time-evolution operator twice and setting , we have:

And since are eigenstates of , we have:

Thus the time-evolution operator can be written as:

Then, acting on a state vector , we have:

Looking at the sum, we can see that the time-evolution operator acts on the state vector by multiplying each eigenstate by a phase factor .

If the state were to be an energy eigenstate , then the time-evolution operator would simply give:

Hence, the state vector would simply pick up a phase factor . It will still be an eigenstate of the Hamiltonian.

Multiple Compatible Observables​

Now suppose we have multiple compatible observables where and they are all compatible with the Hamiltonian . This means that any eigenstate of one observable is also an eigenstate of the other observables. We can denote this eigenstate as or just for short. Performing the same analysis as above, we have:

Stationary States and Expectation Values​

A stationary state is a state where the expectation value of the observable does not change in time. This means that the expectation value of the observable is constant in time. We can show that energy eigenkets are stationary states as follows:

Consider an observable that is compatible with the Hamiltonian . At , suppose the state of the system is an eigenket of . Now let be another observable that is not compatible with . Then, the expectation value of at time is given by:

This means that the expectation value of is constant in time.

Now we look at non-stationary states. This time, the initial state of the system is not an eigenstate of the Hamiltonian. Instead, it is a superposition of energy eigenstates:

Then, the expectation value of the observable at time is given by:

So we get a sum of terms that oscillate in time. The frequency of the oscillation is known as Bohr's frequency condition:

Schrödinger Wave Equation​

The Schrödinger equation can also be written in the form of a wave equation. Suppose we take both sides of the Schrödinger equation and take an inner product with a position eigenket . This gives

We can rewrite the left-hand side as

For the right-hand side, we use a specific form of the Hamiltonian, such as

for a particle in a potential . Then, the right-hand side becomes

Putting both sides together, we have the Schrödinger wave equation, which states

This equation describes how the position-space wavefunction evolves in time under the influence of a potential . Although in the state vector formalism, this is just a specific representation of the Schrödinger equation, it is often used as a starting point in introductory quantum mechanics courses. The mathematics using this as a starting point is often referred to as wave mechanics.

If the Hamiltonian is time-independent, then the wavefunction is

If we plug this into the Schrödinger wave equation , we get

where is the position-space wavefunction of the energy eigenstate . This is Schrödinger's time-independent wave equation.

Summary and Next Steps​

In this section, we began discussing how quantum systems evolve in time.

Here are the key points to remember:

  • Quantum systems evolve in time, described by different pictures (Schrödinger, Heisenberg, and path integral).

  • The time-evolution operator is a unitary operator that describes how the state of the system evolves in time. By enforcing the properties of the time-evolution operator, we derived the Schrödinger equation:

  • We focus on the case of time-independent Hamiltonians, where the time-evolution operator can be expressed as:

  • Generally, the approach for a time-independent Hamiltonian is to:

    1. Find an operator that commutes with the Hamiltonian .
    2. Find the eigenstates of . These are energy eigenkets.
    3. Given an initial state , express it in terms of the eigenkets of .
    4. Apply the time-evolution operator to the state vector .

  • A stationary state is a state where the expectation value of the observable does not change in time. All energy eigenkets are stationary states.

  • Non-stationary states are superpositions of energy eigenstates. They oscillate in time, and the frequency of the oscillation is given by Bohr's frequency condition:

  • The Schrödinger equation can also be expressed in the form of a wave equation, which describes how the position-space wavefunction evolves in time under the influence of a potential. It is given by:

    If the Hamiltonian is time-independent, then we have the time-independent Schrödinger wave equation:

In the next page, we will explore some examples of physical systems and apply the Schrödinger equation to them.

References​

  • J.J. Sakurai, "Modern Quantum Mechanics", Section 2.1 ("Time Evolution and the Schrödinger Equation")