Chapter 12, Adiabatic principle Video Solutions, The Physics of Quantum Mechanics

Problem 1

We have derived approximate expressions for the change in the energies of stationary states when an electric or magnetic field is applied. Discuss whether the derivation of these results implicitly assumed the validity of the adiabatic principle.

Amita Prajapat

Numerade Educator

02:55

Problem 2

In $\S 12.2$ we assumed that the potential energy of air molecules is infinitely large inside a bicycle pump's walls. This cannot be strictly true. Give a reasoned order-of-magnitude estimate for the potential in the walls, and consider how valid it is to approximate this by infinity.

Kayla Day

Numerade Educator

04:44

Problem 3

Explain why $E / \omega$ is an adiabatic invariant of a simple harmonic oscillator, where $\omega$ is the oscillator's angular frequency. Einstein proved this result in classical physics when he was developing the "old quantum theory", which involved quantising adiabatic invariants such as $E / \omega$ and angular momentum. Derive the result for a classical oscillator by adapting the derivation of the WKBJ approximation to the oscillator's equation of motion $\ddot{x}=-\omega^{2} x$

Ajay Singhal

Numerade Educator

01:52

Problem 4

Consider a particle that is trapped in a one-dimensional potential well $V(x)$. If the particle is in a sufficiently highly excited state of this well, its typical de Broglie wavelength may be sufficiently smaller than the characteristic lengthscale of the well for the WKBJ approximation to be valid. Explain why it is plausible that in this case
$$
\frac{1}{\hbar} \int_{x_{1}}^{x_{2}} \mathrm{~d} x^{\prime} \sqrt{2 m\left\{E-V\left(x^{\prime}\right)\right\}}=n \pi
$$
where the $E-V\left(x_{i}\right)=0$ and $n$ is an integer. Relate this condition to the quantisation rule $\oint \mathrm{d} x p_{x}=n h$ used in the "old quantum theory".

Keshav Singh

Numerade Educator

02:28

Problem 5

Show that the "old quantum theory" (Problem 12.4) predicts that the energy levels of the harmonic oscillator are $n \hbar \omega$ rather than $\left(n+\frac{1}{2}\right) \hbar \omega$. Comment on the dependence on $n$ of the fractional error in $E_{n}$.

Keshav Singh

Numerade Educator

12:36

Problem 6

Suppose the charge carried by a proton gradually decayed from its current value, $e$, being at a general time $f e$. Write down an expression for the binding energy of a hydrogen atom in terms of $f$. As $\alpha \rightarrow 0$ the binding energy vanishes. Explain physically where the energy required to free the electron has come from.
When the spring constant of an oscillator is adiabatically weakened by a factor $f^{4}$, the oscillator's energy reduces by a factor $f^{2}$. Where has the energy gone?
In Problems $3.14$ and $3.15$ we considered an oscillator in its ground state when the spring constant was suddenly weakened by a factor $f=$ $1 / 16$. We found that the energy decreased from $\frac{1}{2} \hbar \omega$ to $0.2656 \hbar \omega$ not to $\hbar \omega / 512$. Explain physically the difference between the sudden and adiabatic cases.

Ali Mazrui

Numerade Educator

03:36

Problem 7

Photons are trapped inside a cavity that has perfectly reflecting walls which slowly recede, increasing the cavity's volume $\mathcal{V}$. Give a physical motivation for the assumption that each photon's frequency $\nu \propto \mathcal{V}^{-1 / 3}$. Using this assumption, show that the energy density of photons $u \propto \mathcal{V}^{-4 / 3}$ and hence determine the scaling with $\mathcal{V}$ of the pressure exerted by the photons on the container's walls.
Black-body radiation comprises an infinite set of thermally excited harmonic oscillators - each normal mode of a large cavity corresponds to a new oscillator. Initially the cavity is filled with black-body radiation of temperature $T_{0}$. Show that as the cavity expands, the radiation continues to be black-body radiation although its temperature falls as $\mathcal{V}^{-1 / 3}$. Hint: use equation $(6.125)$.

Chai Santi

Numerade Educator

03:32

Problem 8

Show that when a charged particle gyrates with energy $E$ in a uniform magnetic field of flux density $B$, the magnetic moment $\mu=$ $E / B$ is invariant when $B$ is changed slowly. Hint: recall Problem 9.6. By applying the principle that energy must be conserved when the magnetic field is slowly ramped up, deduce whether a plasma of free electrons forms a para- or diamagnetic medium.

Sandro Maludze

Numerade Educator