# Harmonic Oscillator Spectrum and Eigenstates

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The one-dimensional harmonic oscillator consists of a particle moving under the influence of a harmonic oscillator potential, which has the form, where is the "spring constant".

We see that as Therefore, all stationary states of this system are bound, and thus the energy spectrum is discrete and non-degenerate. Furthermore, because the potential is an even function, the parity operator commutes with Hamiltonian, and thus the wave functions will be either even or odd.

The energy spectrum and the energy eigenstates can be found by either an algebraic method using raising and lowering operators, which is described below, or by solving the Schrödinger equation for the system, as described in the next section.

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## Solution of the Harmonic Oscillator by Operator Methods

The Hamiltonian of the one-dimensional harmonic oscillator is:

or, in terms of the natural frequency,

With the aid of the operator identity,

we may factorize the Hamiltonian as follows.

If we now define the operators,

and

we may write the Hamiltonian as

One may easily show that the operators and satisfy the commutation relation,

Let us now define the Hermitian operator, We denote the (normalized) eigenstate of associated with the eigenvalue as i.e., Note that any eigenstate of is also an eigenstate of the Hamiltonian, with eigenvalue

One may verify that the eigenvalue by acting on the left of this definition with There is therefore a lower bound of on the energy of any state of the harmonic oscillator. This "zero-point energy" is a remarkable and significant feature peculiar to quantum mechanics. One may view this as a consequence of the Heisenberg uncertainty principle; because it is impossible to perfectly localize a particle in both position and momentum spaces, a particle in a harmonic oscillator potential will always possess a non-zero energy relative to the minimum of the potential.

We may now determine what the operators and do to the eigenstates of Acting to the left on the definition, with each of these operators and employing the above commutation relation, we may show that

and

We have thus shown that is a "lowering operator", in the sense that, when applied to the eigenstate of with eigenvalue we obtain a result that is proportional to the eigenstate with eigenvalue For a similar reason, is a "raising operator".

We may use the above results to further restrict the possible values of We may show that it is quantized, and can only take non-negative integer values; i.e., Let us suppose that an eigenstate of with a positive non-integer eigenvalue exists. Without loss of generality, let us suppose that since one can generate such a state from any other such eigenstate by repeated application of the lowering operator. If we act on this state with the lowering operator , then we will generate an eigenstate with a *negative* eigenvalue (note that the factor, does not vanish in this case!), which cannot exist, as pointed out earlier.

The only way to guarantee that no states with negative values of are generated is to restrict to be a non-negative integer. This is guaranteed because, by repeated application of the lowering operator, we will eventually obtain the state, and Therefore, we have shown that is the ground state of the harmonic oscillator.

So, starting from any energy eigenstate, we can construct all other energy eigenstates by applying or repeatedly. In particular, by repeated application of the raising operator, we may generate all of the eigenstates of the harmonic oscillator from its ground state:

## The Ground State Wave Function

We may use the above results to easily determine the ground state of the harmonic oscillator in position space. Starting from the fact that we may write, remembering that in the position basis,

This is a first-order ordinary differential equation, which can easily be solved; the solution is

where is a normalization constant. Upon normalization, we find that the ground state wave function is

We may obtain the ground state wave function in momentum space as well. Remembering that, in this case, the differential equation satisfied by the momentum-space wave function is

and its normalized solution is

## The Excited State Wave Functions

Given the ground state wave function, we may obtain the excited state wave functions by repeated application of as described earlier. In position space,

We may show that

where is a Hermite polynomial.

In the momentum representation, the excited states may be written as

or

Note the appearance of the imaginary unit that is not present in the position representation of these states. This solution can also be obtained via a Fourier transformation of the position representation wave functions.

Notice that there are two factors in the wave functions for the excited states, a Gaussian function and a Hermite polynomial. The former causes the wave function to decrease exponentially as as required of any bound state, while the later accounts for the behavior of the wave function at short distances and the number of nodes of the wave function.

The quantum harmonic oscillator is of particular interest as a problem due to the fact that it can be used to (at least approximately) describe many different systems. A few examples include molecular vibrations, quantum LC circuits, and phonons in solids.

## Problems

**(1)** Calculate the expectation value of the position in an eigenstate of the harmonic oscillator.

**(2)** Calculate the expectation value of the momentum in an eigenstate of the harmonic oscillator.

**(3)** Show that the average kinetic energy, is equal to the average potential energy, This is a special case of the virial theorem, which we will discuss in a later section.

(See Liboff, Richard *Introductory Quantum Mechanics*, 4th Edition, Problem 7.10 for reference.)