Chapter 7 The Harmonic Oscillator

7.1 Why Study the Harmonic Oscillator?

7.2 Review of the Classical Oscillator

7.3 Quantization of the Oscillator (Coordinate Basis)

Exercise 7.3.1 Consider the question why we tried a power-series solution for Eq. (7.3.11) but not Eq. (7.3.8). By feeding in a series into the latter, verify that a three-term recursion relation between $C_{n+2}$, $C_{n}$, and $C_{n-2}$ obtains, from which the solution does not follow so readily. The problem is that $\psi^{\prime\prime}$ has two powers of $y$ less than $2 \varepsilon \psi$, while the $-y^{2}$ piece has two more powers of $y$. In Eq. (7.3.11) on the other hand, of the three pieces $u^{\prime\prime}$, $-2yu^{\prime}$, and $(2\varepsilon-1)u$, the last two have the same powers of $y$.

Exercise 7.3.2 Verify that $H_{3}(y)$ and $H_{4}(y)$ obey the recursion relation, Eq. (7.3.15).

Exercise 7.3.3 If $\psi(x)$ is even and $\phi(x)$ is odd under $x \rightarrow-x$, show that

Use this to show that $\psi_2(x)$ and $\psi_1(x)$ are orthogonal. Using the values of Gaussian integrals in Appendix A.2 verify that $\psi_2(x)$ and $\psi_0(x)$ are orthogonal.

Exercise 7.3.4 Using Eqs. (7.3.23)-(7.3.25), show that

Exercise 7.3.5 Using the symmetry arguments from Exercise 7.3.3 show that $\langle n|X|n\rangle=\langle n|P|n\rangle=0$ and thus that $\left\langle X^2\right\rangle=(\Delta X)^2$ and $\left\langle P^2\right\rangle=(\Delta P)^2$ in these states. Show that $\langle 1| X^2|1\rangle=3 \hbar / 2 m \omega$ and $\langle 1| P^2|1\rangle=\dfrac{3}{2} m \omega \hbar$. Show that $\psi_0(x)$ saturates the uncertainty bound $\Delta X \cdot \Delta P \geqslant \hbar / 2$.

Exercise 7.3.6 Consider a particle in a potential

What are the boundary conditions on the wave functions now? Find the eigenvalues and eigenfunctions.

Exercise 7.3.7 (The Oscillator in Momentum Space) By setting up an eigenvalue equation for the oscillator in the $P$ basis and comparing it to Eq. (7.3.2), show that the momentum space eigenfunctions may be obtained from the ones in coordinate space through the substitution $x \rightarrow p$, $m \omega \rightarrow 1 / m \omega$. Thus, for example,

There are several other pairs, such as $\Delta X$ and $\Delta P$ in the state $|n\rangle$, which are related by the substitution $m \omega \rightarrow 1 / m \omega$. You may wish to watch out for them. (Refer back to Exercise 7.3.5.)

7.4 The Oscillator in the Energy Basis

Exercise 7.4.1 Compute the matrix elements of $X$ and $P$ in the $|n\rangle$ basis and compare with the result from Exercise 7.3.4.

Exercise 7.4.2 Find $\langle X\rangle$, $\langle P\rangle$, $\left\langle X^2\right\rangle$, $\left\langle P^2\right\rangle$, $\Delta X \cdot \Delta P$ in the state $|n\rangle$.

Exercise 7.4.3 (Virial Theorem) The virial theorem in classical mechanics states that for a particle bound by a potential $V(r)=a r^k$, the average (over the orbit) kinetic and potential energies are related by

when $c(k)$ depends only on $k$. Show that $c(k)=k / 2$ by considering a circular orbit. Using the results from the previous exercise show that for the oscillator $(k=2)$

in the quantum state $|n\rangle$.

Exercise 7.4.4 Show that $\langle n\mid X^{4}\mid n\rangle=(\hbar/2m\omega)^2[3+6n(n+1)]$.

Exercise 7.4.5 At $t=0$ a particle starts out in $|\psi(0)\rangle=1/2^{1/2}(0\rangle+|1\rangle)$.

(1)Find $|\psi(t)\rangle$;

(2)Find $\langle X(0)\rangle=\langle\psi(0)| X|\psi(0)\rangle,\langle P(0)\rangle,\langle X(t)\rangle,\langle P(t)\rangle$;

(3)Find $\langle\dot{X}(t)\rangle$ and $\langle\dot{P}(t)\rangle$ using Ehrenfest's theorem and solve for $\langle X(t)\rangle$ and $\langle P(t)\rangle$ and compare with part (2).

Exercise 7.4.6 Show that $\langle a(t)\rangle=\mathrm{e}^{-\mathrm{i}\omega t}\langle a(0)\rangle$ and that $\left\langle a^{\dagger}(t)\right\rangle=\mathrm{e}^{\mathrm{i} \omega t}\left\langle a^{\dagger}(0)\right\rangle$.

Exercise 7.4.7 Verify Eq. (7.4.40) for the case

(1)$\Omega=X$, $\Lambda=X^{2}+P^{2}$.

(2)$\Omega=X^{2}$, $\Lambda=P^{2}$.

The second case illustrates the ordering ambiguity.

Exercise 7.4.8 Consider the three angular momentum variables in classical mechanics:

(1)Construct $L_{x}$, $L_{y}$, and $L_{z}$, the quantum counterparts, and note that there are no ordering ambiguities.

(2)Verify that $\left\{l_{x}, l_{y}\right\}=l_{z}$ [see Eq. (2.7.3) for the definition of the Poisson Bracket].

(3)Verify that $\left[L_{x}, L_{y}\right]=\mathrm{i} \hbar L_{z}$.

Exercise 7.4.9 Consider the unconventional (but fully acceptable) operator choice

in the $X$ basis.

(1)Verify that the canonical commutation relation is satisfied.

(2)It is possible to interpret the change in the operator assignment as a result of a unitary change of the $X$ basis:

where

First verify that

i.e.,

Next verify that

i.e.,

Exercise 7.4.10 Recall that we always quantize a system by promoting the Cartesian coordinates $x_1, \ldots, x_N$; and momenta $p_1, \ldots, p_N$ to operators obeying the canonical commutation rules. If non-Cartesian coordinates seem more natural in some cases, such as the eigenvalue problem of a Hamiltonian with spherical symmetry, we first set up the differential equation in Cartesian coordinates and then change to spherical coordinates (Section 4.2). In Section 4.2 it was pointed out that if $\mathscr{H}$ is written in terms of non-Cartesian but canonical coordinates $q_1 \ldots q_N$; $p_1 \ldots p_N$; $\mathscr{H}\left(q_i \rightarrow q_i, p_i \rightarrow-\mathrm{i}\hbar \partial / \partial q_i\right)$ does not generate the correct Hamiltonian $H$, even though the operator assignment satisfies the canonical commutation rules. In this section we revisit this problem in order to explain some of the subtleties arising in the direct quantization of non-Cartesian coordinates without the use of Cartesian coordinates in intermediate stages.

(1) Consider a particle in two dimensions with

which leads to

in the coordinate basis. Since the problem has rotational symmetry we use polar coordinates

in terms of which

Since $\rho$ and $\phi$ are not mixed up as $x$ and $y$ are [in the $\left(x^2+y^2\right)^{1 / 2}$ term] the polar version can be more readily solved.

The question we address is the following: why not start with $\mathscr{H}$ expressed in terms of polar coordinates and the conjugate momenta

(where $\mathbf{e}_\rho$ is the unit vector in the radial direction), and

i.e.,

and directly promote all classical variables $\rho$, $p_\rho$, $\phi$, and $p_\phi$ to quantum operators obeying the canonical commutations rules? Let's do it and see what happens. If we choose operators

that obey the commutation rules, we end up with

which disagrees with Eq. (7.4.41). Now this in itself is not serious, for as seen in the last exercise the same physics may be hidden in two different equations. In the present case this isn't true: as we will see, the Hamiltonians in Eqs. (7.4.41) and (7.4.42) do not have the same eigenvalues. (What we will see is that $P_{\rho}=-\mathrm{i}\hbar\mathrm{d}/\mathrm{d}\rho$, and hence the $H$ constructed with it, are non-Hermitian.) We know Eq. (7.4.41) is the correct one, since the quantization procedure in terms of Cartesian coordinates has empirical support. What do we do now?

(2) A way out is suggested by the fact that although the choice $P_\rho \rightarrow -\mathrm{i} \hbar \partial / \partial \rho$ leads to the correct commutation rule, it is not Hermitian! Verify that

(You may assume $\rho \psi_1^* \psi_2 \rightarrow 0$ as $\rho \rightarrow 0$ or $\infty$. The problem comes from the fact that $\rho \mathrm{d} \rho \mathrm{d} \phi$ and not $\mathrm{d} \rho \mathrm{d} \phi$ is the measure for integration.)

Show, however, that

is indeed Hermitian and also satisfies the canonical commutation rule. The angular momentum $P_\phi \rightarrow -\mathrm{i} \hbar \partial / \partial \phi$ is Hermitian, as it stands, on single-valued functions: $\psi(\rho, \phi)=\psi(\rho, \phi+2 \pi)$.

(3) In the Cartesian case we saw that adding an arbitrary $f(x)$ to $-\mathrm{i} \hbar \partial / \partial x$ didn't have any physical effect, whereas here the addition of a function of $\rho$ to $-\mathrm{i} \hbar \partial / \partial \rho$ seems important. Why? [Is $f(x)$ completely arbitrary? Mustn't it be real? Why? Is the same true for the $-\mathrm{i} \hbar / 2 \rho$ piece?]

(4) Feed in the new momentum operator $P_\rho$ and show that

which still disagrees with Eq. (7.4.41). We have satisfied the commutation rules, chosen Hermitian operators, and yet do not get the right quantum Hamiltonian. The key to the mystery lies in the fact that $\mathscr{H}$ doesn't determine $H$ uniquely since terms of order $\hbar$ (or higher) may be present in $H$ but absent in $\mathscr{H}$. While this ambiguity is present even in the Cartesian case, it is resolved by symmetrization in all interesting cases. With non-Cartesian coordinates the ambiguity is more severe. There are ways of constructing $H$ given $\mathscr{H}$ (the path integral formulation suggests one) such that the substitution $P_\rho \rightarrow-i \hbar(\partial / \partial \rho+1 / 2 \rho)$ leads to Eq. (7.4.41). In the present case the quantum Hamiltonian corresponding to

is given by

Notice that the additional term is indeed of nonzero order in $\hbar$.

7.5 Passege from the Energy Basis to the $X$ Basis

Exercise 7.5.1 Project Eq. (7.5.1) on the $P$ basis and obtain $\psi_{0}(p)$.

Exercise 7.5.2 Project the relation

on the $X$ basis and derive the recursion relation

using Eq. (7.3.22).

Exercise 7.5.3 Starting with

and

and Eq. (7.3.22), derive the relation

Exercise 7.5.4 Thermodynamics of Oscillators. The Boltzmann formula

where

gives the probability of finding a system in a state $i$ with energy $E(i)$, when it is in thermal equilibrium with a reservoir of absolute temperature $T=1/\beta k$, $k=1.4\times 10^{-16}~\mathrm{ergs}/^{\circ}\mathrm{K}$; being Boltzmann's constant. (The “probability” referred to above is in relation to a classical ensemble of similar systems and has nothing to do with quantum mechanics.)

(1) Show that the thermal average of the system's energy is

(2) Let the system be a classical oscillator. The index $i$ is now continuous and corresponds to the variables $x$ and $p$ describing the state of the oscillator, i.e.,

and

and

Show that

and that

Note that $E_{\mathrm{cl}}$ is independent of $m$ and $\omega$.

(3) For the quantum oscillator the quantum number $n$ plays the role of the index $i$. Show that

and

(4) It is intuitively clear that as the temperature $T$ increases (and $\beta=1 / k T$ decreases) the oscillator will get more and more excited and eventually (from the correspondence principle)

Verify that this is indeed true and show that “large $T$” means $T \gg \hbar \omega / k$.

(5) Consider a crystal with $N_0$ atoms, which, for small oscillations, is equivalent to $3 N_0$ decoupled oscillators. The mean thermal energy of the crystal $\overline{E}_{\text{crystal}}$ is $\overline{E}_{\text{cl}}$ or $\overline{E}_{\text{qu}}$ summed over all the normal modes. Show that if the oscillators are treated classically, the specific heat per atom is

which is independent of $T$ and the parameters of the oscillators and hence the same for all crystals. (More precisely, for crystals whose atoms behave as point particles with no internal degrees of freedom.) This agrees with experiment at high temperatures but not as $T \rightarrow 0$. Empirically,

Following Einstein, treat the oscillators quantum mechanically, assuming for simplicity that they all have the same frequency $\omega$. Show that

where $\theta_E=\hbar \omega / k$ is called the Einstein temperature and varies from crystal to crystal. Show that

Although $C_{\mathrm{qu}}(T) \rightarrow 0$ as $T \rightarrow 0$, the exponential falloff disagrees with the observed $C(T) \xrightarrow[T \rightarrow 0]{} T^3$ behavior. This discrepancy arises from assuming that the frequencies of all normal modes are equal, which is of course not generally true. [Recall that in the case of two coupled masses we get $\omega_{\mathrm{I}}=(k / m)^{1 / 2}$ and $\omega_{\mathrm{II}}=(3 k / m)^{1 / 2}$.] This discrepancy was eliminated by Debye.