Quantum Simple Harmonic Oscillator

The quantum simple harmonic oscillator is a fundamental model in quantum mechanics that describes the behavior of a particle in a potential well. This is especially useful because many physical systems can be approximated as harmonic oscillators near their equilibrium positions.

Table of Contents

• Table of Contents
• Introduction
  • Creation and Annihilation Operators
• Position, Momentum, and Energy
  • Matrix Elements of Creation and Annihilation Operators
  • Matrix Elements of Position and Momentum Operators
  • Energy Eigenfunctions in Position Space
• Expectation Values
• Uncertainty Relation
• Summary and Next Steps

Introduction

Recall that for a classical harmonic oscillator, simple harmonic motion occurs when a particle is subject to a restoring force proportional to its displacement from equilibrium. More abstractly, when the equation of motion is linear and time-translationally invariant, the solution is a combination of harmonic oscillators. For example, take this equation of motion:

This is a second-order linear ordinary differential equation (ODE) with constant coefficients. It is linear because the coefficients of and its derivatives are constants, and it is time-translationally invariant because the coefficients do not depend on time.

For a simple undamped harmonic oscillator, the equation of motion is:

Its solution is a combination of two sinusoidal functions. We typically use complex exponentials to represent the solution instead, because they are easier to manipulate mathematically.

where and are complex constants, and and are the amplitude and phase of the oscillation, respectively. The energy of the oscillator is given by the sum of its kinetic and potential energy:

where is the momentum of the particle, is its mass, and is the spring constant. In the quantum mechanical case, the Hamiltonian can be obtained by replacing the classical variables with their quantum operator equivalents

In the - representation, the position-space wavefunction is difficult to work with. It is more convenient to introduce two (non-Hermitian) operators, and , called the annihilation and creation operators, respectively. Think of this change as changing the basis of the Hilbert space. An analogy would be changing from a pair of Cartesian coordinates to two complex numbers that are conjugates of each other, :

Their commutator is:

And we can always rewrite the position and momentum operators in terms of the annihilation and creation operators:

Next, we define the number operator as:

This can be evaluated as:

The Hamiltonian can thus be expressed in terms of the number operator as:

which is a much simpler form. Since is linear in , they share the same eigenstates. Recall that the eigenstates of are the energy eigenstates. We shall denote the eigenstates of as ; . When we apply to the eigenstates of , we get:

Hence, the energy eigenvalues are:

When , we have the ground state energy . As we will see later, we must have be a non-negative integer. If that is the case, then the ground state is the lowest energy state of the system. Furthermore, the energy eigenvalues are equally spaced by . This means that the energy levels are quantized, and the oscillator can only occupy discrete energy states.

Creation and Annihilation Operators

As we have seen, and are non-Hermitian operators that act on the energy eigenstates of the harmonic oscillator. We call the annihilation operator and the creation operator. To see why they are named as such, we can apply them to the energy eigenstates.

First, the commutation relations between , , and are:

Recall that ; so the eigenvalue of is the energy level of the state. Now suppose we take :

In other words, is an eigenstate of with eigenvalue . This means that is the state with one more quantum of energy than . Hence, "creates" a quantum of energy in the system when it acts on the state .

Similarly, we can take :

This means that is an eigenstate of with eigenvalue . As such, is the state with one less quantum of energy than . This means that "annihilates" a quantum of energy in the system when it acts on the state .

Because both and are eigenstates of with eigenvalue , they must be scalar multiples of each other. We can write this as:

where is a constant. We set to be real and positive by convention. To find , we can "square" both sides of the equation by multiplying both sides by their duals:

Since is , the left-hand side is simply . The right-hand side is . Thus, we have:

And since is real and positive, we have:

Similarly, for the creation operator, we can write:

Thus we have the following relations:

Because , we can split the right-hand side into two terms:

is thus the inner product of two states. Because inner products are positive-definite, we can conclude that . Next, suppose we take a state and repeatedly apply the annihilation operator to it:

If is an integer, we can keep applying the annihilation operator until we reach . Then, applying the annihilation operator again gives:

Meaning that the state is completely destroyed by the annihilation operator. However, if is not an integer, we can keep applying the annihilation operator indefinitely, until we reach a state with negative energy. This contradicts the fact that . Thus, we must have be a non-negative integer.

Position, Momentum, and Energy

We shall first write down all the possible eigenstates of the system, . Let be the ground state of the system. Then, we can repeatedly apply the creation operator to it to get the other eigenstates:

Thus:

Matrix Elements of Creation and Annihilation Operators

The matrix elements of the creation and annihilation operators are easy to compute.

For example, for the annihilation operator, we start with Equation and apply another state to it:

Recall that eigenstates are orthogonal to each other. Therefore, if , we have:

If , we have . Thus we can summarize the matrix elements of the annihilation operator as:

Similarly, we can compute the matrix elements of the creation operator. Begin with Equation and apply another state to it:

Matrix Elements of Position and Momentum Operators

Using the matrix elements of the creation and annihilation operators, as well as Equations and , we can compute the matrix elements of the position and momentum operators. For the position operator, we have:

Similarly, for the momentum operator, we have:

Energy Eigenfunctions in Position Space

Recall that for a continuous observable, the eigenstates of the observable are represented by wavefunctions in position space. In our case, this wavefunction is known as the energy eigenfunction. The energy eigenfunctions are the wavefunctions of the harmonic oscillator in position space:

This is actually quite tricky to compute. Start with the ground state , defined by . Obviously, applying any operator or product of operators to the ground state will still yield zero. Applying a position to it gives:

Recall that in the position representation, the momentum operator is given by:

Thus, we can rewrite the equation as:

Dividing both sides by gives:

Let (where the square root is taken for dimensionality reasons). Then we have:

Rearranging this, and using , gives:

This is a separable differential equation;

Then, integrating both sides gives:

To normalize the wavefunction, we can integrate over all space. I will skip over the details (it's just a Gaussian integral), but the result is:

The subsequent eigenstates can be computed by applying the creation operator to the ground state. For example,

Recall that the creation operator is given by

so we have

For the general case , we have:

We can also express the energy eigenfunctions in terms of Hermite polynomials. The Hermite polynomials are defined as

The Hermite polynomials are orthogonal polynomials, meaning that they are orthogonal to each other with respect to the weight function . In other words,

Then, the energy eigenfunctions can be expressed as:

Expectation Values

We now compute the expectation values of the position and momentum operators in the energy eigenstates.

We know that is

so the expectation value of is:

The first term is zero because , and . The second term is also zero because , and . The third term is

and the last term is

Thus we have

In the ground state this is just . The expectation value of is zero from Equation . The expectation value of is also zero from Equation .

We can also compute the expectation value of . Recall that is given by

so the expectation value is

which, in the ground state, is just .

Lastly, for the kinetic and potential energies, we have

Thus, the expectation values of the kinetic and potential energies are equal, and they are both equal to half of the total energy.

Uncertainty Relation

We conclude this section with the uncertainty relation. First, the uncertainty in position squared is given by

which, in the ground state, is .

The uncertainty in momentum squared is given by

which, in the ground state, is .

Multiplying these two uncertainties gives:

In the ground state, this reduces to , just like in the Gaussian wave packet. This is expected because the ground state has a Gaussian shape.

Notice that as increases, the product of the uncertainties increases. This also obeys the Heisenberg uncertainty principle, as expected.

Summary and Next Steps

We have derived many important results relating to the quantum harmonic oscillator. We have shown that the energy eigenvalues are quantized and equally spaced, and we have derived the energy eigenfunctions in position space.

Below is a summary of the results we have derived:

• The Hamiltonian of the quantum harmonic oscillator is given by:

• The creation and annihilation operators are given by:

  And they obey the commutation relation . The Hamiltonian can be expressed in terms of the creation and annihilation operators as:

• The number operator is given by:

• The energy eigenvalues are the eigenvalues of the number operator:

  They are evenly spaced, and the ground state energy is . The creation and annihilation operators act on the energy eigenstates as

  In position-space, the energy eigenfunctions are

  where . Using Hermite polynomials, we can also express the energy eigenfunctions as

  where

• The matrix elements of the various operators are given by:

• The expectation values of the various operators are given by:

• The uncertainty relation is given by:

In the next section, we will discuss the time-evolution of the quantum harmonic oscillator.