Coupled Harmonic Oscillators

We begin our study of quantum field theory with a nonrelativistic system of coupled harmonic oscillators. We have already studied this system as a classical system in the previous section. Now, we apply quantization to this system.

Table of Contents

• Table of Contents
• Uncoupled Harmonic Oscillators
• Two Coupled Harmonic Oscillators
  • Hamiltonian for Two Coupled Oscillators
• Occupation Number Representations

Uncoupled Harmonic Oscillators

In modern particle physics, it is common knowledge that particles can be created and destroyed. For example, an electron and positron can annihilate to create a pair of photons. This means that our state vector turns from to . To take another example, a neutron can decay into a proton, electron, and antineutrino. The state vector turns from to .

This means that we must abandon single-particle wavefunctions. Instead, we take inspiration from the quantum harmonic oscillator and its creation and annihilation operators and . We can think of these operators as the creation and annihilation of particles.

We shall now consider a system of uncoupled harmonic oscillators. In this case, "uncoupled" simply means that the oscillators do not interact with each other. The Hamiltonian for a single harmonic oscillator is given by:

The Hamiltonian for a system of uncoupled harmonic oscillators is just the sum of the Hamiltonians for each oscillator:

Next, we can promote the creation and annihilation operators to act on the entire system of oscillators. The creation operator creates a quanta of energy in the -th oscillator. The annihilation operator destroys a quanta of energy in the -th oscillator.

They act on the system as:

and obey the following commutation relations:

The vacuum state is defined as the state with no quanta in any of the oscillators:

And since the oscillators are independent of each other, we can easily rewrite the Hamiltonian in terms of the creation and annihilation operators:

We can now write a state vector in terms of the vacuum state and the creation and annihilation operators. Recall that for a single oscillator, we have

We can just extend this to a system of oscillators (once again assuming they are uncoupled):

Two Coupled Harmonic Oscillators

We shall now consider two coupled harmonic oscillators. The first oscillator has a distance and the second has a distance . Their state vectors live in the Hilbert spaces and , respectively. They also each have their own position, momentum, creation, and annihilation operators.

Recall that in a coupled state, the overall state vector exists in the tensor product of the two Hilbert spaces:

Each position ket and is a vector in the Hilbert space of the respective oscillator. The combined state vector is given by:

To upgrade our operators to the coupled case, we can just take the tensor product of the two operators:

This works because tensor products on sums act by . We can verify this by checking the action of the operators on the state vector. For example, acting on the state vector gives

Indeed, we get the correct eigenvalue .

Next, we can show that the two position operators commute with each other:

This means that the two positions are compatible observables, and we can hence measure them simultaneously. Skipping some details, here are all the commutation relations for the two oscillators:

Hamiltonian for Two Coupled Oscillators

To write the Hamiltonian for two coupled oscillators, we start with the classical Hamiltonian and apply first quantization. The classical Hamiltonian for two coupled oscillators is given by:

where we have the kinetic energy terms for each oscillator, the potential energy terms for each oscillator, and the coupling term .

We can just replace and with their corresponding operators to get the Hamiltonian in terms of the operators:

(in our notation, since we don't use hats for operators, this looks the same as the classical Hamiltonian. But they are indeed different.)

Recall that when we analyzed the classical system, we found that the normal modes of the system were

1. The center of mass mode, where both oscillators move in the same direction with the same amplitude.
2. The relative mode, where both oscillators move in opposite directions with the same amplitude.

Their vectors are and . We now define the quantum operator analogues of these normal modes:

where we have defined and to be the center of mass and relative coordinates, respectively. We also included a factor of to normalize the operators. Similarly, the momentum operators are defined as:

We can now rewrite the Hamiltonian in terms of these new operators:

The key is that we no longer have the coupling term in the Hamiltonian. This is because we have already diagonalized the Hamiltonian by changing to the normal mode coordinates.

The commutation relations for the new operators are given by:

which are the same as the original commutation relations. We can also define the creation and annihilation operators for the new normal modes:

where and are the frequencies of the normal modes. Then, the Hamiltonian is just

which is the same as the Hamiltonian for two uncoupled harmonic oscillators. Notice that with a change of coordinates, we have made the Hamiltonian diagonal. This means that the two normal modes are independent of each other, just like the uncoupled oscillators.

Occupation Number Representations

In quantum field theory, we replace the single-particle wavefunctions with occupation number representations, where we use creation and annihilation operators to create and destroy particles. Instead of a single-particle wavefunction , we have a state vector , where is the number of particles in the -th state. Next, instead of the creation/annihilation operators acting to create/destroy a quantum of energy, we have the creation/annihilation operators acting to create/destroy a particle in the -th state.

In the following discussion we will be using natural units where .

First, suppose we have a simple particle-in-a-box system, where the solutions to the Schrödinger equation in the position space are given by

as well as the eigenstates of the momentum operator. In a system like this, we have boundary conditions; for one, because the wavefunction should be periodic. This means that , entailing .

This means that must be an integer multiple of , or , where is an integer. Thus, the position-space wavefunction is given by

Next, suppose we introduce more particles in the box. Suppose they are interacting bosons. The state vector follows the notation

Applying the momentum operator and the Hamiltonian operator to the state vector gives

where is the energy of the -th particle.

We now introduce the new notation for the state vector. Consider the fact that given a number of particles, we write the state vector as , which is to be read as "two particles (A and C) with momentum and one particle (B) with momentum ". However, given that the labels are arbitrary, and particles with the same momentum are indistinguishable, we can just count the number of particles with each momentum. Thus the state vector becomes , where the first index is the number of particles with momentum , the second index is the number of particles with momentum , and so on. This is called the occupation number representation. We often just shorten it to .