Some More Simple Systems

Abstract

A number of simple quantum systems are considered. Their energy eigenvalue problem is solved and their general behavior is discussed. In particular, the one-dimensional harmonic oscillator, the ammonia molecule and two-state systems in general, the one-dimensional lattice and periodic potentials.

Problems and Exercises

7.1

Consider the system of a quantum particle of mass m moving in one dimension and subject to a harmonic force $F=-m\omega ^2x$. The system is initially ($t=0$) in the state with wave function $\psi (x,0)\,=\,\Theta (x)\,\psi _1(x)$, where $\psi _1(x)$ is the energy eigenfunction of the first excited state. Calculate the probability to find the system in the ground state at any time $t>0$.

7.2

Find the energy eigenfunctions and eigenvalues for the system of a particle in the potential

$$V(x)\,=\,\left\{ \begin{array}{cc} +\infty \,&{}\,(x<0)\\ \,&{}\,\\ \frac{1}{2}m\omega ^2 x^2\,&{}\,(x>0) \end{array}\right. $$

7.3

A particle of mass m moves in one dimension subject to a potential

$$V(x)\,=\,\frac{1}{2}m\omega ^2x^2-\lambda x\,,$$

where $\lambda$ and $\omega$ are known parameters.

(a) Find the energy eigenstates and eigenvalues of the system.

(b) Assume that initially ($t=0$) the system occupies the state with wave function

$$\psi (x,0)\,=\,\left( \frac{m\omega }{\hbar \pi }\right) ^{1/4}e^{-\frac{m\omega }{2\hbar }x^2}\,.$$

Calculate the probability to find the particle in the state of lowest energy.

7.4

A harmonic oscillator ($H=p^2/2m+m\omega ^2x^2/2$) is initially ($t=0$) in the state

$$|\psi (0)\rangle \,=\,\frac{1}{\sqrt{2}}\left( |0\rangle \,+\,e^{i\alpha }|1\rangle \,\right) \,,$$

where $\alpha$ is a real parameter and $|n\rangle$ the energy eigenstates. Calculate the expectation values

$$\langle x\rangle _t,\,\,\langle p\rangle _t,\,\,\langle x^2\rangle _t,\,\,\langle p^2\rangle _t$$

and verify Heisenberg’s inequality for the position and momentum at any time.

7.5

Calculate the following commutators of Heisenberg operators for a harmonic oscillator:

$$\left[ \,\hat{x}(t),\,\hat{x}(t')\,\right] ,\,\,\,\left[ \,\hat{p}(t),\,\hat{p}(t')\,\right] ,\,\,\,\left[ \,\hat{x}(t),\,\hat{p}(t')\,\right] \,.$$

7.6

Calculate the quantity

$$\langle 1|e^{\frac{i}{\hbar }\lambda \,\hat{p}}|1\rangle $$

for a harmonic oscillator. $|1\rangle$ is the first excited energy eigenstate and $\lambda$ a real parameter. (Hint: You may use the operator identity $e^{A+B}=e^Ae^Be^{-\frac{1}{2}[A,B]}$, valid whenever the commutator [A, B] is a number.)

7.7

A two-state system, in the framework of an orthonormal basis $|1\rangle ,\,|2\rangle$, has a Hamiltonian matrix

$$\mathcal{{H}}\,=\,\left( \begin{array}{cc} 0\,&{}\,E_0\\ \,&{}\,\\ E_0\,&{}\,0 \end{array}\right) \,,$$

with $E_0$ a real parameter. The initial state of the system is

$$|\psi (t)\rangle \,=\,\frac{1}{\sqrt{2}}\left( |1\rangle \,+\,e^{i\alpha }|2\rangle \,\right) $$

with $\alpha$ a real parameter. Find the probability to return in this state as a function of the time $t>0$.

7.8

Prove that a harmonic oscillator ($H=p^2/2m+m\omega ^2x^2/2$) in an arbitrary state will have an uncertainty $(\Delta x)_t^2$ which will be a periodic function of time with period $(2\omega )^{-1}$.

7.9

Consider a harmonic oscillator ($H=p^2/2m+m\omega ^2x^2/2$) in the state

$$|\psi \rangle \,=\,e^{\frac{i}{\hbar }\lambda \hat{p}}|n\rangle \,,$$

where $|n\rangle$ is an energy eigenstate and $\lambda$ is a real parameter. Calculate the uncertainties in position and momentum and verify the Heisenberg inequality.

7.10

Consider a two-state system characterized by the three physical observables $\hat{A},\,\hat{B},\,\hat{C}$. For a given orthonormal basis $|1\rangle ,\,|2\rangle$ the corresponding operators are represented by the matrices

$$\mathcal{{A}}=\left( \begin{array}{cc} 0&{}1\\ \,&{}\,\\ 1&{}0 \end{array}\right) ,\,\,\,\,\,\mathcal{{B}}\,=\,\left( \begin{array}{cc} 0&{}-i\\ \,&{}\,\\ i&{}0 \end{array}\right) ,\,\,\,\,\,\mathcal{{C}}\,=\,\left( \begin{array}{cc}1\,&{}\,0\\ \,&{}\,\\ 0\,&{}\,-1 \end{array}\right) .$$

(a) Find the eigenstates and eigenvalues of $\hat{A}$ and $\hat{B}$.

(b) Assume that a measurement of $\hat{C}$ is performed and the outcome is the value $+1$. Immediately afterward, the system is subject to a measurement of $\hat{B}$. What is the probability of an eigenvalue $+1$? Assume that the outcome of the second measurement is the eigenvalue $+1$. Immediately afterward, the system is subject to a third measurement of $\hat{C}$ again. What is the probability of an eigenvalue $-1$?

7.11

A particle of mass m moves in one dimension subject to the potential

$$V(x)\,=\,-\frac{\hbar }{2m}W'(x)+\frac{W^2(x)}{2m}\,.$$

Verify that there exists a zero energy bound state and write down its wave function for $W(x)=m\omega x+\lambda x^3$ (with $\omega$ and $\lambda$ real parameters).

7.12

For a particle moving in one dimension, the probability amplitude to make a transition from a point x to a point $x'$ in time T is given by

$$\mathcal{{K}}(x',x;T)\,=\,\langle x'|e^{-\frac{i}{\hbar }T\hat{H}}|x\rangle \,.$$

If the particle is subject to a periodic potential $V(x+a)=V(x)$, show that

$$\mathcal{{K}}(x'+a,x+a;T)\,=\,\mathcal{{K}}(x',x;T)\,.$$

Show that the opposite is also true, namely, if $\mathcal{{K}}$ is periodic, the potential will have to be periodic as well.

7.13

For the periodic potential $V(x)\,=\,\sum _{-\infty }^{+\infty }V_0\delta (x-na)$ (Kronig–Penney model) the allowed values of the Bloch wave number k are given in terms of the energy wave number $q=\sqrt{2mE/\hbar ^2}$ by the equation

$$\cos (ka)\,=\,\cos (qa)+\frac{mV_0}{\hbar ^2q}\sin (qa)\,.$$

Show that at the end points of the Brillouin zone $k_n=n\pi /a$ with $n\pi$ very large

$$q(k_n)\,\approx \,\frac{n\pi }{a}\,+\,\epsilon _n$$

with $\epsilon _n<<1$. Determine the possible values of $\epsilon _n$.

7.14

Consider the “double harmonic oscillator”, i.e., a particle of mass m moving in the potential

$$V(x)\,=\,\frac{1}{2}m\omega ^2\left( |x|-a\right) ^2$$

shown in Fig. 7.6.

Fig. 7.6

Double oscillator

You may assume that the distance between the two minima 2a is much larger than the characteristic oscillator length $a>>\sqrt{\hbar /m\omega }$. It is expected that the ground state will have the property $\psi (x)\approx \psi _0(x+a)$ for $x<<0$ and $\psi (x)\approx \psi _0(x-a)$ for $x>>0$, where $\psi _0(x)$ is the ground state wave function of the simple harmonic oscillator. Therefore, a reasonable estimate for the ground state of the system is $\psi (x)\,=\,N\left( \,\psi _0(x+a)\,+\,\psi _0(x-a)\,\right) \,.$ Calculate the expectation value of the energy in this state. Do the same for the state $\tilde{\psi }(x)\,=\,\tilde{N}\left( \,\psi _0(x+a)\,-\psi _0(x-a)\,\right) \,.$ The following approximation is assumed to be valid $\int _0^{\infty }d\xi \,\xi \,e^{-(\xi +a\sqrt{m\omega /\hbar })^2}\,\approx \,\frac{\hbar }{4m\omega a^2}e^{-\frac{m\omega }{\hbar }a^2}\,.$ Hint: You may use the identities $\hat{H}\psi _0(x\pm a)=\frac{\hbar \omega }{2}\psi _0(x\pm a)$ for $x<0$ and $x>0$ correspondingly.

7.15

Consider the simple harmonic oscillator and rewrite the corresponding creation and annihilation operators in terms of the number operator $\hat{N}$ and a “phase” operator $\hat{ \phi }$ as

$$\hat{a}=\left( \hat{N}+1\right) ^{1/2}e^{i\hat{ \phi }},\,\,\,\,\hat{a}=e^{-i\hat{ \phi }}\left( \hat{N}+1\right) ^{1/2}\,.$$

Prove the commutation relations¹⁰

$$\left[ \,e^{i\hat{ \phi }},\,\hat{N}\,\right] =e^{i\hat{ \phi }},\,\left[ e^{-i\hat{ \phi }},\,\hat{N}\right] =-e^{-i\hat{ \phi }},\,[\cos \hat{ \phi },\,\hat{N}]=i\sin \hat{\phi },\,[\sin \hat{ \phi },\,\hat{N}]=-i\cos \hat{ \phi }\,.$$

Finally, show that for any state the following is true:

$$(\Delta N)^2(\Delta \cos \phi )^2\ge \frac{1}{4}\left| \langle \sin \phi \rangle \right| ^2\,.$$