Skip to main content

Time Evolution of Quantum Systems

Up to this point, we have discussed the basic principles of quantum mechanics, including the postulates and mathematical formalism. We have not yet discussed how quantum systems evolve in time. 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, 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.

Postulates of Quantum Mechanics​

With the Schrödinger equation, we have now derived the third postulate of quantum mechanics. We are now ready to partially state all the postulates of quantum mechanics:

  1. State Space: The state of a quantum system is described by a state vector in a Hilbert space. a. The state of a composite system is described by the tensor product of the component systems.
  2. Observables: Observables are represented by Hermitian operators. a. The possible outcomes of a measurement are the eigenvalues of the operator. b. Immediately after a measurement, the state of the system is the eigenvector corresponding to the measured eigenvalue. c. The probability of measuring an eigenvalue is given by the Born rule: . d. The expectation value of an observable is given by the inner product of the state vector and the operator.
  3. Unitary Evolution: The time evolution of a quantum system is governed by the Schrödinger equation. a. The time-evolution operator is a unitary operator. b. The time-evolution operator obeys the Schrödinger equation.

There are, of course, more details to each of these postulates, but this is a good starting point.