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Essential Quantum Mechanics$

Gary Bowman

Print publication date: 2007

Print ISBN-13: 9780199228928

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780199228928.001.0001

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(p.186) Appendix C The Harmonic Oscillator

(p.186) Appendix C The Harmonic Oscillator

Source:
Essential Quantum Mechanics
Publisher:
Oxford University Press

The simple harmonic oscillator (SHO) may be the single most important system in physics. The SHO describes vibrations in classical mechanics, electrical oscillations in an LC circuit, quantum-mechanical vibrations of molecules, and countless other physical systems. Even in quantum electrodynamics—the quantum field theory of the electromagnetic field—the SHO plays a central role, providing a model for photons, the quanta of the field.

The SHO is also one of the few systems for which the time-independent Schrödinger equation (TISE) is exactly soluble. In this appendix, I'll briefly discuss what the SHO is, the SHO solutions to the TISE, and a quite abstract representation of the SHO: the “number representation,” which forms the basis for the SHO's entry into quantum field theory.

C.1 Energy Eigenstates and Eigenvalues

Classically, the SHO is a system subject to a “restoring force” that is linear in displacement from the system's equilibrium point. If we set x = 0 at that point, then the force is given by: F = −kx, where k is a positive constant. (Equivalently, the potential energy is Vkx 2.) The usual physical example of a classical SHO is a mass, m, attached to a spring. Such a system oscillates with a characteristic frequency, denoted ω. The parameters m, ω, and k are related by: ω = k / m .

In quantum mechanics, the SHO can be solved through “standard” means: we form the classical Hamiltonian, and then quantize it, rewriting it in terms of operators. The TISE can be solved exactly for this system, but we'll leave the (non-trivial!) technical aspects aside, focusing instead on results.

The SHO solutions of the TISE are analogous to the energy eigenstates of the infinite potential well (see Chapter 9). That is, they are energy eigenstates, in position representation—wavefunctions corresponding to a specific energy.1

(p.187) For the SHO, the wavefunction boundary conditions are applied at the point where the energy associated with a particular eigenstate equals the oscillator's potential energy. At this point the wavefunction changes from oscillatory to monotonically decreasing—somewhat like what occurs in the step potential and the rectangular barrier (in Chapter 9) for the E < V 0 case.

The SHO is a remarkable system in part because of its energy eigenvalues

(C.1)
E n = ħ ω ( n + 1 2 ) , n = 0 , 1 , 2 ,
Note first that the eigenvalues are uniformly spaced—each separated by ħω from the next. This is a special property, with special implications.

Let's write the nth SHO energy eigenstate as |n〉. Suppose we expand an initial SHO state |α〉 in these eigenstates: |α〉 = ∑j c j|j〉. Chapter 11 then implies that the time evolved state is

(C.2)
| α ( t ) = j e i E j t / ħ c j | j = j e ω t ( j + 1 2 ) c j | j = j [ cos ( ω t ( j + 1 2 ) ) sin ( ω t ( j + 1 2 ) ) ] c j | j .
Equation (C.2)) reveals that |α(t)〉 is periodic, with period T = 4π/ω. But any SHO state can be expanded in the energy representation, so any SHO state is periodic with period T.2 This is a remarkable property of the SHO, arising from its uniformly spaced energy eigenvalues.

There's another interesting feature of the SHO eigenvalues. Our classical, mass-and-spring SHO could start at x = 0 with zero initial velocity. It would then simply remain at rest at x = 0; its energy would be precisely zero.

Even for n = 0, however, the quantum oscillator's energy is non-zero. This zero-point energy is a strictly quantum effect, and it's not unique to the SHO. For example, the energy eigenvalues of the infinite potential well (see Chapter 9) are E n = n 2π2ħ2/2mL 2, with n = 1,2,3,…; the energy can never be zero.

We can at least heuristically justify the zero-point energy by invoking the classical picture of a particle moving along the x axis. Classically, any non-zero energy corresponds to two different possible momenta: one in the +x direction, the other in the −x direction. The quantum-mechanical analog to this situation is an energy eigenstate corresponding to two different (p.188) momenta. This implies an uncertainty in momentum: Δ p x > 0. This picture fails, however, for a state with zero energy. Then there can be only one momentum, 0, so Δ p x = 0.3

Now consider the zero-point energy in terms of the position-momentum uncertainty relation: Δ xΔ p x≥ħ/2. For a system constrained to a finite region of the x axis, as are the SHO and the infinite potential well, 0≤Δ x < ∞. As just argued, a non-zero energy eigenstate satisfies Δ p x > 0, so it's easy to see how the uncertainty relation, also, can be satisfied—that is, how Δ xΔ p x > 0. But a zero-energy eigenstate satisfies Δ p x = 0, and since Δ x is finite, Δ xΔ p x > 0 can't possibly be satisfied.

By this (admittedly non-rigorous) argument, we see that we must have a zero-point energy. Interesting as that fact is, however, the zero-point energy takes on a much deeper significance in the context of quantum electrodynamics.

C.2 The Number Operator and its Cousins

As suggested above, the energy eigenstates and eigenvalues of the SHO can be obtained using “standard” methods. An alternative approach uses what are often called operator, or algebraic, methods. The key to such methods is the introduction of two operators, â and its adjoint â, defined as:

(C.3)
a ^ = m ω 2 ħ ( x ^ + i m ω p ^ x ) , a ^ = m ω 2 ħ ( x ^ i m ω p ^ x ) .
We leave solution of the TISE using â and â to the standard texts. Our concern is the physical meaning of â, â, and another operator, N̂, defined as N̂≡ââ.

Both â and â are non-Hermitian operators,4 and therefore do not represent observable quantities (see Postulate 2, in Chapter 2). However, the SHO Hamiltonian operator Ĥ involves both ◯ and p̂x, as do â and â. It's not too surprising, then, that Ĥ can be written in terms of â and â; explicitly:

(C.4)
H ^ = p ^ x 2 2 m + 1 2 m ω 2 x ^ 2 = ħ ω ( a ^ a ^ + 1 2 ) = ħ ω ( N ^ + 1 2 ) .
Evidently the operator â†â (or N̂) is Hermitian, since Ĥ is Hermitian.5

From Equations (C.1)) and (C.4)), we have

(C.5)
H ^ | n = ħ ω ( N ^ + 1 2 ) | n = E n | n = E n | n = ħ ω ( n + 1 2 ) | n ,
(p.189) where |n〉 is the nth energy eigenstate. Evidently, the action of N̂ on the state |n〉 is to return the number n: N̂|n〉=n|n〉. For this reason, N̂ is called the number operator.

To uncover the meaning of â and â themselves, start with the commutation relation,

(C.6)
[ a ^ , a ^ ] = i ħ ( p ^ x x ^ x ^ p ^ x ) = 1 ,
where Eq. (C.3)) and the commutation relation [◯,p̂x]=iħ have been used. Now consider the commutator
(C.7)
[ N ^ , a ^ ] | ψ = ( a ^ a ^ a ^ a ^ a ^ a ^ ) | ψ = [ a ^ , a ^ ] a ^ | ψ = a ^ | ψ .
(Here |ψ〉 is an arbitrary state, included to clarify the calculation.) Thus, we can write N̂â−âN̂=−â, or N̂â=â(N̂−1).

Now let's act on an SHO energy eigenstate with N̂â

(C.8)
N ^ a ^ | n = a ^ ( N ^ 1 ) | n = a ^ N ^ | n a ^ | n = n a ^ | n a ^ | n = ( n 1 ) a ^ | n .
Rewriting Eq. (C.8)) as N̂[â|n〉]=(n−1)[â|n〉] suggests thinking of â|n〉 as a new state. When the number operator, N̂, acts on â|n〉, it returns the number n−1. Thus, â|n〉=c −| n−1〉. A similar calculation leads to N̂[â|n〉]=(n+1)[â|n〉], so that â|n〉=c +|n+1〉. (Here c and c + are as-yet undetermined constants.)

Up to the constants c + and c , then, â “raises” an SHO energy eigenstate to the next-higher energy eigenstate, and â “lowers” an energy eigenstate to the next-lower energy eigenstate. We thus refer to â and â as raising and lowering operators, respectively.6

Note the similaritites between â and â, on the one hand, and Ĵ and Ĵ+ (of Chapter 8), on the other. All four operators are formed from the sum or difference of Hermitian operators, but the raising and lowering operators themselves are non-Hermitian. And in both cases—angular momentum, or SHO energy—the eigenvalues are uniformly spaced.

Although the raising, lowering, and number operators may seem like curiosities, they can be quite useful in calculations. Moreover, the concepts and methods of the quantum SHO form a cornerstone of quantum electrodynamics, the quantum field theory of electromagnetic interactions.

C.3 Photons as Oscillators

In Newtonian classical mechanics, the fundamental entity is the particle. Only in the late 19th century did fields come to play a central role in (p.190) classical physics, in the form of the classical electromagnetic field. At the same time it became clear that light is a manifestation of electromagnetism: an electromagnetic wave.

In quantum mechanics, the fundamental entity is the quantum state—though we typically think of the state as representing particles (though the meaning of “particle” can be unclear in quantum mechanics). Quantum field theory—a subject well beyond the scope of this book—extends quantum physics to describe not just particles, but also fields.7

In quantum electrodynamics, the entities that comprise the electromagnetic field—the field “quanta”—are called photons.8 We need not delve deeply into the concept of a photon; we require only a simple picture of its role in quantum electrodynamics.

For some physical system, let us call each allowable frequency and direction of electromagnetic wave a mode of the system. In quantum electrodynamics, the quantum field theory of electromagnetic interactions, each photon in a mode of frequency ω contributes an energy ħω to that mode. Compare this to a quantum SHO of frequency ω: because its energy eigenvalues are uniformly spaced in steps of ħω, its energy can only be changed in “chunks” of ħω. This suggests pressing the SHO formalism into service to represent photons.

In quantum electrodynamics, photons are routinely created as the electromagnetic field's energy increases, and annihilated as the field's energy decreases. Then the raising, lowering, and number operators take on physical meaning: â corresponds to creating a photon and â to annihilating a photon; as such, â and â are called creation and annihilation operators, respectively, in quantum electrodynamics. For a state |k〉, the number operator, ââ, again returns k, which now denotes how many photons occupy a particular field mode.

Finally, we return to the SHO's zero-point energy. The similarities between the SHO and photons seem suggestive, but they need not imply that photons actually are oscillators. Still, the correspondence is quite deep.

When the electromagnetic field is represented as photons, the SHO's zero-point energy appears as the vacuum energy—an energy associated with the field even if there are no photons in any of the modes.

Although the vacuum energy—the zero-point energy of the field—has very few observable consequences, it's more than a mere curiosity. In quantum electrodynamics, the vacuum energy leads to a picture of the “vacuum” which is not at all empty, but full of activity, including the continual creation and destruction of virtual particle-antiparticle pairs.

(p.191) To venture further into the vacuum energy, and quantum electrodynamics generally, would lead us into particle physics and quantum field theory proper. These are subjects both wide and deep. If interested, I suggest you consult works that focus on these topics—but be forewarned, it can be very tough going!

Notes:

(1) Detailed discussions of the quantum-mechanical SHO appear in nearly every standard quantum mechanics text. Many, such as Griffiths (2005), Liboff (2003), Schiff (1968), and Townsend (1992), include plots of the first few energy eigenfunctions.

(2) Actually, T is the longest possible period. If, say, c 0 = 0, the period will be less than T.

(3) Please realize that what's crucial is that the spread in p is 0, not that p = 0 per se.

(4) This can be seen by inspection: â≠â, that is, â is not self-adjoint.

(5) The ħω/2 term simply multiplies a state by a constant, and so cannot affect whether Ĥ is Hermitian.

(6) By calculating 〈n|â,â|n〉 and 〈n,â|n〉, the inner product of â|n〉 with itself and the inner product of â|n〉 with itself, respectively, we find that c + = n + 1 and c = n . Thus, we could create “normalized” raising and lowering operators 1 n + 1 a ^ and 1 n a ^ .

(7) In quantum field theory, particles are often thought of as localized excitations of fields.

(8) A photon is not simply a “particle of light.” See Loudon (2000), pp. 1–2.