Chapter 3, Oscillators Video Solutions, The Physics of Quantum Mechanics

Problem 1

After choosing units in which everything, including $\hbar=1$, the Hamiltonian of a harmonic oscillator may be written $H=\frac{1}{2}\left(p^{2}+x^{2}\right)$, where $[x, p]=\mathrm{i}$. Show that if $|\psi\rangle$ is a ket that satisfies $H|\psi\rangle=E|\psi\rangle$, then
$$
\frac{1}{2}\left(p^{2}+x^{2}\right)(x \mp \mathrm{i} p)|\psi\rangle=(E \pm 1)(x \mp \mathrm{i} p)|\psi\rangle
$$
Explain how this algebra enables one to determine the energy eigenvalues of a harmonic oscillator.

Ameer Said

Numerade Educator

07:07

Problem 2

Given that $A\left|E_{n}\right\rangle=\alpha\left|E_{n-1}\right\rangle$ and $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega$, where the annihilation operator of the harmonic oscillator is
$$
A \equiv \frac{m \omega x+\mathrm{i} p}{\sqrt{2 m \hbar \omega}}
$$
show that $\alpha=\sqrt{n}$. Hint: consider $\left.|A| E_{n}\right\rangle\left.\right|^{2}$.

Hafiz Shahzaib

Numerade Educator

02:54

Problem 3

The pendulum of a grandfather clock has a period of $1 \mathrm{~s}$ and makes excursions of $3 \mathrm{~cm}$ either side of dead centre. Given that the bob weighs $0.2 \mathrm{~kg}$, around what value of $n$ would you expect its non-negligible quantum amplitudes to cluster?

Narayan Hari

Numerade Educator

05:32

Problem 4

Show that the minimum value of $E(p, x) \equiv p^{2} / 2 m+\frac{1}{2} m \omega^{2} x^{2}$ with respect to the real numbers $p, x$ when they are constrained to satisfy $x p=\frac{1}{2} \hbar$, is $\frac{1}{2} \hbar \omega$. Explain the physical significance of this result.

Sam Stansfield

Numerade Educator

01:54

Problem 5

How many nodes are there in the wavefunction $\langle x \mid n\rangle$ of the $n^{\text {th }}$ excited state of a harmonic oscillator?

Cathy Geisel

Numerade Educator

06:05

Problem 6

Show that in terms of a harmonic oscillator's characteristic length $\ell \equiv \sqrt{\hbar / 2 m \omega}$ the ladder operators can be written
$$
A=\frac{x}{2 \ell}+\ell \frac{\partial}{\partial x} \quad \text { and } \quad A^{\dagger}=\frac{x}{2 \ell}-\ell \frac{\partial}{\partial x}
$$
Hence show that the wavefunction of the second excited state is $\langle x \mid 2\rangle=$ constant $\times\left(x^{2} / \ell^{2}-1\right) \mathrm{e}^{-x^{2} / 4 \ell^{2}}$ and find the normalising constant.

Ameer Said

Numerade Educator

02:29

Problem 7

Explain why the wavefunction $\langle x \mid n\rangle$ of the oscillator's $n^{\mathrm{th}}$ stationary state must have the form
$$
\langle x \mid n\rangle=H_{n}(x / \ell) \mathrm{e}^{-x^{2} / 4 \ell^{2}}
$$
where $H_{n}$ is an $n^{\text {th }}$-order ('Hermite') polynomial. By casting the equations $A|n\rangle=\sqrt{n}|n-1\rangle$ and $A^{\dagger}|n-1\rangle=\sqrt{n}|n\rangle$ in the $x$-representation, show that
$$
H_{n}^{\prime}(x / \ell)=\sqrt{n H}_{n-1}(x / \ell) \quad \text { and } \quad \sqrt{n H}(x / \ell)=\frac{x}{\ell} H_{n-1}(x / \ell)-H_{n-1}^{\prime}(x / \ell)
$$ and thus that
$$
\sqrt{n} H_{n}(x / \ell)=\frac{x}{\ell} H_{n-1}(x / \ell)-\sqrt{n-1} H_{n-2}(x / \ell)
$$
Given that $H_{0}=\left(2 \pi \ell^{2}\right)^{-1 / 4}$ and $H_{1}(y)=y /\left(2 \pi \ell^{2}\right)^{1 / 4}$, use this recurrence relation to reproduce the plots of the wavefunctions $\langle x \mid 2\rangle$ and $\langle x \mid 40\rangle$ shown in Figure 3.4. Explain the physical significance of the vertical arrows. Why is the amplitude of $\langle x \mid 40\rangle$ largest near the right arrow?

Adriano Chikande

Numerade Educator

07:07

Problem 8

Use
$$
x=\sqrt{\frac{\hbar}{2 m \omega}}\left(A+A^{\dagger}\right)=\ell\left(A+A^{\dagger}\right)
$$
to show for a harmonic oscillator that in the energy representation the operator $x$ is
$$
x_{j k}=\ell\left(\begin{array}{cccccccc}
0 & \sqrt{1} & 0 & 0 & \ldots & & & & \\
\sqrt{1} & 0 & \sqrt{2} & 0 & & & & & \\
0 & \sqrt{2} & 0 & \sqrt{3} & \cdots & & & & \\
& & \sqrt{3} & \cdots & & & & & \\
\ldots & & \ldots & & \cdots & & \cdots & & \\
& & & \cdots & 0 & \sqrt{n-1} & \ldots & & \\
& & & & \sqrt{n-1} & 0 & \sqrt{n} & & \\
& & & & & \sqrt{n} & 0 & \sqrt{n+1} & \ldots \\
& & & & \ldots & & \sqrt{n+1} & 0 & \\
\ldots & & \ldots & & \ldots & & \ldots & & \ldots
\end{array}\right)
$$
Calculate the same entries for the matrix $p_{j k}$.

Hafiz Shahzaib

Numerade Educator

04:20

Problem 9

Show that the momentum operator of a harmonic oscillator can be expressed in terms of the creation and annihilation operators as
$$
p=\frac{\mathrm{i} \hbar}{2 \ell}\left(A^{\dagger}-A\right) \quad \text { where } \quad \ell \equiv \sqrt{\frac{\hbar}{2 m \omega}}
$$
Hence show that
$$
\left\langle 0\left|p^{2}\right| 0\right\rangle=\left(\frac{\hbar}{2 \ell}\right)^{2}
$$
How does this result relate to the physics of a free particle discussed in $\$ 233 ?

Hafiz Shahzaib

Numerade Educator

05:33

Problem 10

At $t=0$ the state of a harmonic oscillator, mass $m$, frequency $\omega$, is
$$
|\psi\rangle=\frac{1}{\sqrt{2}}|N-1\rangle+\frac{1}{\sqrt{2}}|N\rangle
$$
Show that subsequently
$$
\langle x\rangle_{t}=\sqrt{N} \ell \cos (\omega t) \quad \text { where } \quad \ell \equiv \sqrt{\frac{\hbar}{2 m \omega}}
$$
Interpret this result physically. What does this example teach us about the validity of classical mechanics?

Show that a classical oscillator with energy $\left(N+\frac{1}{2}\right) \hbar \omega$ has amplitude
$$
x_{\max }=2 \sqrt{N+\frac{1}{2}} \ell
$$
To explain the discrepancy between these results, consider the case in which initially
$$
|\psi\rangle=\frac{1}{\sqrt{K}} \sum_{k=N}^{N+K-1}|k\rangle
$$
with $N \gg K \gg 1$. Show that then $\langle x\rangle_{t} \simeq 2 \sqrt{N} \ell \cos (\omega t)$ consistent with classical physics.

Khaled Yasein

Numerade Educator

03:47

Problem 11

By expressing the annihilation operator $A$ of the harmonic oscillator in the momentum representation, obtain $\langle p \mid 0\rangle$. Check that your expression agrees with that obtained from the Fourier transform of
$$
\langle x \mid 0\rangle=\frac{1}{\left(2 \pi \ell^{2}\right)^{1 / 4}} \mathrm{e}^{-x^{2} / 4 \ell^{2}}, \quad \text { where } \quad \ell \equiv \sqrt{\frac{\hbar}{2 m \omega}}
$$

Lottie Adams

Numerade Educator

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Problem 12

Show that for any two $N \times N$ matrices $A, B, \operatorname{trace}([A, B])=0$. Comment on this result in the light of the results of Problem $3.8$ and the canonical commutation relation $[x, p]=\mathrm{i} \hbar$.

Victor Salazar

Numerade Educator

17:54

Problem 13

A Fermi oscillator has Hamiltonian $H=f^{\dagger} f$, where $f$ is an operator that satisfies
$$
f^{2}=0, \quad f f^{\dagger}+f^{\dagger} f=1
$$
Show that $H^{2}=H$, and thus find the eigenvalues of $H$. If the ket $|0\rangle$ satisfies $H|0\rangle=0$ with $\langle 0 \mid 0\rangle=1$, what are the kets (a) $|a\rangle \equiv f|0\rangle$, and (b) $|b\rangle \equiv f^{\dagger}|0\rangle ?$
In quantum field theory the vacuum is pictured as an assembly of oscillators, one for each possible value of the momentum of each particle type. A boson is an excitation of a harmonic oscillator, while a fermion in an excitation of a Fermi oscillator. Explain the connection between the spectrum of $f^{\dagger} f$ and the Pauli principle.

Mahnoor Amin

Numerade Educator

21:28

Problem 14

$\mathrm{In}$ the time interval $(t+\delta t, t)$ the Hamiltonian $H$ of some system varies in such a way that $|H| \psi\rangle \mid$ remains finite. Show that under these circumstances $|\psi\rangle$ is a continuous function of time.
A harmonic oscillator with frequency $\omega$ is in its ground state when the stiffness of the spring is instantaneously reduced by a factor $f^{4}<$ 1, so its natural frequency becomes $f^{2} \omega .$ What is the probability that the oscillator is subsequently found to have energy $\frac{3}{2} \hbar f^{2} \omega ?$ Discuss the classical analogue of this problem.

Mirza Aslam Beig

Numerade Educator

01:39

Problem 15

$P$ is the probability that at the end of the experiment described in Problem 3.14, the oscillator is in its second excited state. Show that when $f=\frac{1}{2}, P=0.144$ as follows. First show that the annihilation operator of the original oscillator
$$
A=\frac{1}{2}\left\{\left(f^{-1}+f\right) A^{\prime}+\left(f^{-1}-f\right) A^{\prime \dagger}\right\}
$$
where $A^{\prime}$ and $A^{\prime \dagger}$ are the annihilation and creation operators of the final oscillator. Then writing the ground-state ket of the original oscillator as a sum $|0\rangle=\sum_{n} c_{n}\left|n^{\prime}\right\rangle$ over the energy eigenkets of the final oscillator, show that the condition $A|0\rangle=0$ yields the recurrence relation
$$
c_{n+1}=-\frac{f^{-1}-f}{f^{-1}+f} \sqrt{\frac{n}{n+1}} c_{n-1}
$$
Finally using the normalisation of $|0\rangle$, show numerically that $c_{2} \simeq 0.3795$. What value do you get for the probability of the oscillator remaining in the ground state?
Show that at the end of the experiment the expectation value of the energy is $0.2656 \hbar \omega$. Explain physically why this is less than the original ground-state energy $\frac{1}{2} \hbar \omega$.
This example contains the physics behind the inflationary origin of the universe: gravity explosively enlarges the vacuum, which is an infinite collection of harmonic oscillators (Problem 3.13). Excitations of these oscillators correspond to elementary particles. Before inflation the vacuum is unexcited so every oscillator is in its ground state. At the end of inflation, there is non-negligible probability of many oscillators being excited and each excitation implies the existence of a newly created particle.

Manik Pulyani

Numerade Educator

09:05

Problem 16

In terms of the usual ladder operators $A, A^{\dagger}$, a Hamiltonian can be written
$$
H=\mu A^{\dagger} A+\lambda\left(A+A^{\dagger}\right)
$$
What restrictions on the values of the numbers $\mu$ and $\lambda$ follow from the requirement for $H$ to be Hermitian?
Show that for a suitably chosen operator $B, H$ can be rewritten
$$
H=\mu B^{\dagger} B+\text { constant }
$$
where $\left[B, B^{\dagger}\right]=1 .$ Hence determine the spectrum of $H$.

Abhijit Das

Numerade Educator

01:39

Problem 17

Numerically calculate the spectrum of the anharmonic oscillator shown in Figure $3.2$. From it estimate the period at a sequence of energies. Compare your quantum results with the equivalent classical results.

Suzanne W.

Numerade Educator

06:58

Problem 18

Let $B=c A+s A^{\dagger}$, where $c \equiv \cosh \theta, s \equiv \sinh \theta$ with $\theta$ a real constant and $A, A^{\dagger}$ are the usual ladder operators. Show that $\left[B, B^{\dagger}\right]=1$. Consider the Hamiltonian
$$
H=\epsilon A^{\dagger} A+\frac{1}{2} \lambda\left(A^{\dagger} A^{\dagger}+A A\right)
$$
where $\epsilon$ and $\lambda$ are real and such that $\epsilon>\lambda>0 .$ Show that when
$$
\epsilon c-\lambda s=E c, \quad \lambda c-\epsilon s=E s
$$
with $E$ a constant, $[B, H]=E B .$ Hence determine the spectrum of $H$ in terms of $\epsilon$ and $\lambda$.

Abhijit Das

Numerade Educator