The Harmonic Oscillator

• Many physical systems involve oscillations about an equilibrium point. If the amplitude of the oscillations is small, the potential energy function (in that vicinity) is approximately a parabola, and we have simple harmonic motion.

• If we put the coordinate origin at the equilibrium point, and choose E = 0 to be the value of U(x) at the equilibrium point, then we can write

• We will examine the stationary states for this potential energy. We assume the total energy to be small enough so that the particle is really bound. (Of course, if the above expression described U everywhere, all energies would give bound states, but this can only be an approximation of some more realistic potential energy function.)

• It is customary to introduce the angular frequency of the oscillation of the (classical) particle, and write w² = k/m.

• The TISE becomes

• We will look first for the ground state. We learned from the square well that the lowest energy state is the one for which the wavefunction has no zeros in the (classically) allowed region. (This state has the longest wavelength, lowest frequency, hence lowest energy.) Also, the solutions must be either even or odd functions of x (because of the nature of the equation) and hence the lowest state solution must be an even function. Finally, the solution must die away in the forbidden region.

• These considerations lead us to guess:

where A and a are constants.

• Direct substitution shows that this solution works, provided that

• The resulting energy is

• Note again that this lowest energy is not zero.

• The constant A is determined by the normalizations requirement:

The result is that

Excited States of the Oscillator

• Guessing the wavefunction worked for the ground state, but for the higher energy states one must be more systematic.

• An ingenious method involves "factoring" the left side of the TISE. We define two differential operators:

• It is straightforward to show that, applied to any differentiable function of x,

That is, the left side applied to any f(x) gives the same result as the right side applied to the same function.

• This means that we can rewrite the TISE for the oscillator in the form

• These operators do not obey the commutative rule, since

• Thus they obey the "commutation relation"

• Now if we apply a_- to the ground state wavefunction we find

• Calling the ground state wavefunction F₀, we write this as

• Next we use the commutation relation to show that

• This means that the function (a₊F₀) will satisfy the TISE if the energy is

• The operator a₊ has turned the ground state wavefunction into that of the first excited state. It is called a "raising" operator.

• We can see what the wavefunction of this excited state is:

• If we apply a₊ again, we get the second excited state. Since

we see that (a₊a₊F₀) is the wavefunction (apart from a normalization constant) corresponding to energy

• Byt the same procedure, the nth excited state is obtained by n applications of the raising operator:

• It corresponds to energy

• The operator a_- is a "lowering" operator, which moves us downward in the hierarchy of states. For example, start from the first excited state, with wavefunction (apart from a constant) a₊F₀, and apply a_-. The commutaion relation gives

since a_- "annihilates" F₀. Thus a_-, operating on the nth state, gives (apart from a constant) the (n-1)th state.

• There are many practical applications of the quantum oscillator described here. The vibrational motion of atoms in molecules, the thermal vibrations of atoms in a solid, and the quantization of electromagnetic radiation are a few examples.

Observables and Operators

• We return now to general principles. We have discussed the meaning of the wavefunction in terms of the probability of finding the particle at a particular location. The position of the particle is, of course, one quantity we might imagine measuring experimentally. It is an observable quantity.

• But there are many physical observables. One is the energy, which we determine from the solutions of the TISE. That equation has the formal structure

where H denotes the operator



This is called the Hamiltonian (named for a 19th century Irish scientist).

• An equation of the form

is called an eigenvalue equation. The values of "Number" for which an acceptable "Function" exists satisfying this equation are called the eigenvalues, and the corresponding functions are called the eigenfunctions.
 
• The TISE is such an equation. The allowed values of the energy are the eigenvalues of the Hamiltonian operator, and the corresponding wavefunctions are its eigenfunctions.

• This is an example of a general structure in quantum mechanics. The observable quantity (energy) is represented by an operator (the Hamiltonian). The allowed values of the observable are the eigenvalues of the operator, each corresponding to a function (the eigenfunction) which represents the state of the system when the observable has that value.

Probability Distributions

• Because the S-E is a linear equation, any superposition of solutions is also a solution. For example, consider two different states of definite energy (stationary states), with energies E₁ and E₂. Their complete wavefunctions (including the time dependence) are

Each of these obeys the S-E:

• But any superposition such as

also satisfies the S-E, and thus represents a possible state of the system.

• What does this kind of state mean? It has two values of the energy at once. If we measure the energy, what do we find?

• Recall that wavefunctions must obey the normalization condition

• We will assume that both Y₁ and Y₂ obey this condition. Then for Y to obey it requires that

• Now the eigenfunctions of an observable operator corresponding to different eigenvalues are orthogonal. Two functions are orthogonal if

• It follows that the last two integrals above vanish, and we thus require that

• We interpret this as follows: The probability that a measurement of energy will yield the value E₁ is the squared magnitude of C₁, and correspondingly for E₂ and C₂.

• More generally, we postulate that:


The wavefunction of any state can be written as a superposition of eigenfunctions of any observable operator, in the form



where f_n are the eigenfunctions corresponding to the eigenvalue l_n of the operator. The eigenfunctions are normalized and orthogonal to each other. Then the probability that a measurement of the observable will yield the value l_n is C_n*C_n.

• A corollary is that if we make a number of measurements of the observable on systems, all described by the same wavefunction, then the average of the measurements will be

• This average value can also be written in the more direct form:

where O_l is the operator representing the observable.

• Consider as an example the superposition, discussed above, of the two energy eigenfunctions. The average value of the energy in that state is

Here H is the Hamiltonian operator.
 
• These interpretive postulates concerning the eigenfunctions and eigenvalues of observable operators, and the interpretation of the coefficients in the expansion of the system's wavefunction in a series of those eigenfunctions, is the central part of the Copenhagen interpretation of quantum mechanics.

• Clearly it is important to know what the operators are for physically important observables. In classical physics those observables are always some functions of the position and momentum of the particle.

• The operator for position (x) is simple: multiplication by x. Thus, for example, the average position of the particle, when its state is described by Y(x,t), is given by

• We consider next the operator for the momentum. Using these two operators one can construct (or guess at) the operators for other observables that appear in classical physics, such as angular momentum.

• Later we will see that there are some observables that are peculiar to the microscopic world, having no classical counterpart.

The Momentum Operator

• A completely free particle has the wavefunction

• This corresponds to a definite value of the momentum (p). It is therefore an eigenfunction of the momentum operator. What is that operator?

• We note that

• This shows that the momentum operator is

• We can use this to construct the kinetic energy operator. Classically E_k = p²/2m, so we expect the operator to be

• We recognize this as the first term in the Hamiltonian. In fact, the Hamiltonian is constructed by adding the kinetic energy operator to the potential energy operator. The latter is simply multiplication by the potential energy function U(x).

• Since the eigenfunctions of momentum are the exponentials , the expansion of a general wavefunction in terms of these functions is exactly the same as a Fourier transform:

The constant in front of the integral is there so that

• The function f(p,t) plays the role of the expansion coefficients in the discussion above. That is, the probability that the particle's momentum will be found to lie between p and p+dp is given by

• The squared magnitude of f thus gives the probability distribution for momentum, just as the squared magnitude of Y gives the probability distribution for position.

• It is of some importance that the position operator and momentum operator do not commute. Applied to any function of x, we have the operator equation

• The physical implications of this include the Uncertainty Principle.