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The HET Group

HOMEWORK 4
TEXTBOOKS:

Quantum Mechanics, Vol. 2, by Claude CohenTannoudji, et al.,
WileyVCH.
Quantum Mechanics with Basic Field Theory, by Bipin R. Desai, Cambridge.
Quantum Mechanics, by Eugen Merzbacher, Hamilton.
Modern Quantum Mechanics, by J. J. Sakurai, AddisonWesley.

due date: Tue., April 19, 2011
Problem 4.1
Consider a onedimensional harmonic oscillator of mass $m$, angular frequency $\omega_0$ and charge $q$.
Let $n\rangle$ and $E_n = (n+ \frac{1}{2} ) \hbar\omega_0$ be the eigenstates and corresponding eigenvalues of its Hamiltonian $H_0$.
For $t\lt 0$, the oscillator is in the ground state $0\rangle$.
At $t=0$, it is subjected to an electric field pulse of duration $\tau$.
The corresponding perturbation can be written as
\[ W(t) = \left\{ \begin{array}{ccl} q\mathcal{E} x & , & 0\le t \le \tau \\
0 & , & t\lt 0 ~~~\mathrm{or}~~~ t\gt \tau \end{array} \right. \]
Let $\mathcal{P}_{0n}$ be the probability of finding the oscillator in the state $n\rangle$ after the pulse.
 Calculate $\mathcal{P}_{01}$ using firstorder timedependent perturbation
theory. How does it vary with $\tau$ for fixed $\omega_0$?
 Calculate $\mathcal{P}_{02}$ using secondorder timedependent perturbation theory.
 Find the exact eigenvalues of the Hamiltonian $H_0+W$ and corresponding eigenstates (you may find it helpful to use the translation operator $e^{iap/\hbar}$, where $a$ is a length parameter to be determined).
Deduce exact expressions for $\mathcal{P}_{01}$ and $\mathcal{P}_{02}$ (no need to sum the series). By expanding in $\mathcal{E}$, confirm the results of parts a. and b.
Problem 4.2
Consider two spin 1/2 particles coupled by an interaction of the form
\[ W = a(t) \vec S_1 \cdot \vec S_2 \]
where $a(t)\to 0$ as $t\to \pm\infty$ and the integral
\[ \mathcal{A} = \int_{\infty}^{+\infty} a(t) dt \]
is finite.
 If at $t=\infty$ the system is in the state $+\rangle$, calculate the state of the system at $t=+\infty$.
Find the probability $\mathcal{P} (+ \longrightarrow +)$ of finding the system in the state $+\rangle$ at $t=+\infty$ in terms of $\mathcal{A}$.
 Calculate $\mathcal{P} (+\longrightarrow +)$ using firstorder timedependent perturbation theory.
Compare your result to the exact expression found in part a. and deduce the conditions for the validity of the approximation required for firstorder perturbation theory.
 Now assume that the two spinors are also interacting with a uniform mangetic field in the $z$direction, $\vec B = B_0 \hat z$.
The corresponding Hamiltonian is
\[ H_0 =  B_0 ( \gamma_1 S_{1z} + \gamma_2 S_{2z} ) \]
where $\gamma_1 \ne \gamma_2$.
Assuming
\[ a(t) = a_0 e^{t^2/\tau^2} \]
calculate $\mathcal{P} (+\longrightarrow +)$ using firstorder timedependent perturbation theory. Plot this probability as a function of $B_0$ for fixed $a_0$ and $\tau$.
Problem 4.3
A simple model of the photoelectric effect
Consider a particle of mass $m$ and charge $q$ constrained to move along the $x$axis under the influence of the potential
\[ V(x) =  \alpha \delta(x) \ \ , \ \ \ \ \alpha \gt 0 \]
 Show that there is a single bound state of (negative) energy
\[ E_0 =  \frac{m\alpha^2}{2\hbar^2} \]
with corresponding eigenfunction
\[ \phi_0 (x) = \frac{\sqrt{m\alpha}}{\hbar}\ e^{m\alpha x/\hbar^2} \]
and that for each positive energy
\[ E = \frac{\hbar^2k^2}{2m} \gt 0 \]
there are two stationary states corresponding to a particle incident from the left and right, respectively.
Show that the eigenfunction for a particle coming from the left is
\[ \chi_k (x) = \left\{ \begin{array}{lll} \frac{1}{\sqrt{2\pi}}
\left[ e^{ikx}  \frac{1}{1+\frac{i\hbar^2 k}{m\alpha}} e^{ikx} \right]
& , & x\lt 0 \\
\frac{1}{\sqrt{2\pi}} \frac{\frac{i\hbar^2 k}{m\alpha}}{1 + \frac{i\hbar^2 k}{m\alpha}} e^{ikx} & , & x\gt 0 \end{array} \right. \]
where the normalization is chosen so that (show!)
\[ \langle \chi_k  \chi_{k'} \rangle = \delta (kk') \]
HINT: insert a factor $e^{\epsilon x}$ where $\epsilon\gt 0$ and let $\epsilon\to 0$ at the end of the calculation. Use (with proof)
\[ \lim_{\epsilon\to 0^+} \frac{\epsilon}{\epsilon^2 + q^2} = \pi \delta (q) \]
 Calculate the matrix element of the position operator
\[ \langle \chi_k  X  \phi_0 \rangle \]
 The particle interacts with an oscillating electric field of angular frequency $\omega$ where $\hbar\omega \gt E_0$. The interaction Hamiltonian is
\[ W(t) = q\mathcal{E} x \sin (\omega t) \]
If the particle is initially in the bound state $\phi_0\rangle$,
what is the transition probability $w$ per unit time to a positive energy state of energy $E\gt 0\ $? (photoelectric or photoionization effect)
How does $w$ vary with $\omega$ and $\mathcal{E}\ $?
Problem 4.4
Random perturbation  simple relaxation model
A physical system is described by the Hamiltonian $H_0$. At time $t=0$ it is in the eigenstate $i\rangle$ ($H_0i\rangle = E_i i\rangle$).
The system is subject to a perturbation $W(t)$. Let $\mathcal{P}_{if} (t)$ be the probability of finding the system in the eigenstate $f\rangle$ at time $t$ ($H_0 f\rangle = E_f f\rangle$). The transition probability per unit time is
\[ w_{if} (t) = \frac{d \mathcal{P}_{if} (t)}{dt} \]
 Using firstorder perturbation theory show that
\[ w_{if} (t) = \frac{1}{\hbar^2} \int_0^t d\tau\ e^{i\omega_{fi}\tau} W_{fi} (t) W_{fi}^* (t\tau) + \mathrm{c.c.} \]
where $\hbar\omega_{fi} = E_f  E_i$.
 Consider a very large number $N$ of systems which are identical but do not interact with each other.
Each system has a different microscopic environment and consequently sees a different perturbation $W^{(k)} (t)$ ($k=1,2,\dots ,N$).
It is impossible to know with certainty each perturbation $W^{(k)} (t)$ and can only specify statistical averages,
\[ \overline{W_{fi} (t)} = \frac{1}{N} \sum_{k=1}^N W_{fi}^{(k)} (t) \ \ , \ \ \ \
\overline{W_{fi} (t) W_{fi}^* (t\tau)} = \frac{1}{N} \sum_{k=1}^N W_{fi}^{(k)} (t) W_{fi}^{(k)*} (t\tau) \]
This perturbation is said to be random.
A random perturbation is called stationary if the statistical averages are time independent.
Then the unperturbed Hamiltonian $H_0$ is redefined so that all
\[ \overline{W_{fi}} = 0 \]
We define the correlation function of the perturbation by
\[ g_{fi} (\tau) = \overline{W_{fi} (t) W_{fi}^* (t\tau)} \]
It generally vanishes for $\tau \gg \tau_c$ where $\tau_c$ is the correlation time of the perturbation.
Thus the perturbation has a "memory" that extends into the past over a time interval of order $\tau_c$.
 Suppose that the $N$ systems are all in the state $i\rangle$ at $t=0$ and are subject to a stationary random perturbation of correlation function $g_{fi} (\tau)$ and correlation time $\tau_c$.
Calculate the proportion $\pi_{if} (t)$ of these systems that go into the state $f\rangle$ per unit time.
Show that after a certain time ($t\gg t_1$, with $t_1$ to be determined), $\pi_{if} (t)$ is independent of time.
 Let
\[ g_{fi} (\tau) = v_{fi}^2 e^{\tau /\tau_c} \]
Find the constant $\pi_{if}$ for $t\gg t_1$ and plot it as a function of $\omega_{fi}$ for fixed $\tau_c$.
 The above results are valid only for $t\ll t_2$, since they are derived using perturbation theory. Estimate $t_2$.
Since $t_1 \ll t_2$ (why?), find the condition for having a transition probability per unit time which is independent of $t$ for $g_{fi}$ given in part ii.
 Application to a simple system.
Suppose that each of the $N$ systems is a spin 1/2 particle of gyromagnetic ratio $\gamma$ placed in a uniform field in the $z$direction, $\vec B = B_0\hat z$ and set $\omega_0 = \gamma B_0$.
These spinors are enclosed in a spherical cell of radius $R$ bouncing constantly back and forth between the walls.
Let $t_v$ be the flight time (mean time between two collisions of a spinor with the spherical wall).
Between collisions, the spinor sees only the field $\vec B$. During a collision with the wall, a spinor remains absorbed on the surface for a mean time $\tau_a \ll \tau_v$
during which it sees a microscopic magnetic field $\vec b$ in addition to $\vec B$ due to the paramagnetic impurities in the wall.
The direction of $\vec b$ varies randomly from one collision to another. Let $b_0$ be the mean value of $\vec b$.
 What is the correlation time of the perturbation seen by the spinors?
Assuming a correlation function of the form given in part b.ii., show that
\[ \overline{b_i (t) b_j (t\tau)} = \left\{ \begin{array}{ccc} \frac{1}{3} b_0^2 \frac{\tau_a}{\tau_v} e^{\tau/\tau_a} & , & i=j \\
0 & , & i\ne j \end{array} \right. \]
where $i,j=x,y,z$.
 Let $\vec M\ $ be the macroscopic magnetization of the $N$ spinors.
Show that, under the effect of the collisions with the wall, $M_z$ (the component of $\vec M\ $ along $\vec B$) relaxes with a time constant (longitudinal relaxation time) $T_1$,
\[ \frac{dM_z}{dt} =  \frac{M_z}{T_1} \]
and find $T_1$.
 Show that by studying the variation of $T_1$ with $B_0$ one can determine the absorption time $\tau_a$ experimentally.
 Suppose we are given several cells of different radii $R$ constructed from the same material.
By measuring $T_1$, how can we determine experimentally the mean amplitude $b_0$ of the microscopic field at the wall?
Problem 4.5
Absorption of radiation by a Hydrogen atom
Consider two particles of masses $m_1$ and $m_2$ and opposite charges $q_1$ and $q_2$ (electron and proton in the case of a Hydrogen atom).
Let $\vec r_i$, $\vec p_i$ ($i=1,2$) be their position vectors and momenta.
Define
\[ \vec r = \vec r_1  \vec r_2 \ \ , \ \ \ \ \vec r_G = \frac{m_1 \vec r_1 + m_2 \vec r_2}{m_1+m_2} \ \ , \ \ \ \ \vec p = \frac{m_2\vec p_1  m_1 \vec p_2}{m_1+m_2} \ \ , \ \ \ \ \vec p_G = \vec p_1 + \vec p_2 \]
and also let
\[ M = m_1 + m_2 \ \ , \ \ \ \ m = \frac{m_1m_2}{m_1+m_2} \ \ , \ \ \ \ q_1 = q_2 = q \]
 Show that the Hamiltonian can be split as
\[ H_0 = H_{ext} (\vec r_G, \vec p_G) + H_{int} (\vec r , \vec p) \]
Show that the eigenstates of $H_{ext}$ are $\vec K\rangle$ with corresponding eigenvalue $\frac{\hbar^2 K^2}{2M}$,
\[ H_{ext} \vec K\rangle = \frac{\hbar^2 K^2}{2M} \vec K\rangle \]
What are the bound states of $H_{int}$?
 Consider two eigenstates of $H_{int}$, $a\rangle$ and $b\rangle$ of energies $E_a$ and $E_b$, respectively. Suppose $E_b\gt E_a$ and let
\[ E_b  E_a = \hbar\omega_0 \]
What energy must be supplied to the atom to move it from the state $\vec K ; a\rangle$ (i.e., the atom in the state $a\rangle$ with a total momentum $\hbar\vec K$) to the state $\vec K' ; b\rangle$?
 The atom interacts with a plane electromagnetic wave of wave vector $\vec k$ and angular frequency $\omega = ck$. The vector potential is
\[ \vec A ( \vec r, t) = A_0 \vec e \ e^{i (\vec k\cdot \vec r \omega t)} + \mathrm{c.c.} \]
where $\vec e \perp \vec k$ is the polarization vector.
The principal term of the interaction Hamiltonian between the electromagnetic wave and the atom is
\[ W(t) =  \frac{q_1}{m_1} \vec p_1 \cdot \vec A  \frac{q_2}{m_2} \vec p_2 \cdot \vec A \]
Express $W(t)$ in terms of $\vec r$, $\vec p$, $\vec r_G$, $\vec p_G$, $m$, $M$ and $q$.
Show that in the electric dipole approximation,
\[ \vec k \cdot \vec r \ll 1 \]
we have
\[ W(t) = W e^{i\omega t} + \mathrm{c.c.} \ \ , \ \ \ \ W =  \frac{qA_0}{m} \vec e \cdot \vec p \ e^{i\vec k \cdot \vec r_G} \]
 Find a relation between $\vec K$, $\vec K'$ and $\vec k$ in order to have a nonvanishing matrix element
\[ \langle \vec K' ; bW (t) \vec K ; a \rangle \]
Interpret this relation in terms of momentum conservation during the absorption of an incident photon by the atom.
 Show that if an atom in the state $\vec K ; a\rangle$ is placed in the above electromagnetic wave, resonance occurs when the energy $\hbar\omega$ of the photons differs from the energy $\hbar\omega_0$ of the atomic transition $a\rangle \to b\rangle$ by a quantity $\delta$ to be determined (since $\delta$ is small, you may replace $\omega$ by $\omega_0$ in the expression for $\delta$).
Write $\delta$ as
\[ \delta = \delta_1 + \delta_2 \]
where $\delta_1$ depends on $\vec K$ and the angle between $\vec K$ and $\vec k$ (Doppler effect) and $\delta_2$ is independent of $\vec K$.
Show that $\delta_2$ is the recoil kinetic energy of an atom initially at rest that absorbs a resonant photon.
Show that $\delta_2 \ll \delta_1$ when $\hbar\omega_0 \simeq 10$ eV (the domain of atomic physics). Use $M\simeq 10^9$ eV (mass of proton) and for $K$ a value that corresponds to a thermal velocity at temperature $T = 300$ K.
Would $\delta_2$ be negligible compared to $\delta_1$ if $\hbar\omega_0 \simeq 10^5$ eV (the domain of nuclear physics)?
