Physics 309 Final

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Questions (3 pts. apiece) Answer questions in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions circle the correct answer.

1. If the hydrogen atom wave function is ${ \vert nlm \rangle}= R_{nl}(r) Y_l^m(\theta,\phi)$, then what is the definition of $\langle nlm\vert nlm\rangle$? In other words, what is the inner product for the hydrogen atom eigenfunctions in terms of $R_{nl}$, $Y_l^m(\theta,\phi)$, $r$, $\theta$, $\phi$, constants, etc.? What is this inner product equal to?

   
   
   
   
   
   
   

2. Consider the density plot for the $n=8, l=3, m=1$ state shown below. What, if anything, does this density plot tell you about the trajectories of the electrons in the hydrogen atom?

   

   
   
   

3. Recall the vibration-rotation spectrum of carbon monoxide shown in the figure. The peaks are separated by constant energy $\hbar^2/I$ except at the center of the spectrum where the separation is twice that value (the `gap'). Why?

   

4. List one experimental result that led to the development of quantum mechanics. Why was that result important?

   
   
   
   
   
   
   

5. Why is quantum mechanics a better theory for explaining solar fusion than classical physics? You don't have to calculate anything to answer this question.

   
   
   
   
   
   
   

6. Consider a particle in a box confined to the range $x = 0-2~\rm\AA$ with an initial wave function $\psi(x,t=0)$ shown in the figure. What happens to the wave function for $t>0$? Your answer should be descriptive and qualitative, not quantitative.

   

   
   

7. A hydrogen atom is in the initial state
   
   $$\vert \psi(r,\theta,\phi,t=0) \rangle = \frac{\vert 321\rangle + 3\vert 211\rangle}{\sqrt{10}}$$
   
   
   with energy eigenstates $E_n = -13.6~eV/n^2$. What is the solution to the Schroedinger equation for $T>0$ in terms $n$, $E_n$, the elements of $\vert\psi(r,\theta,\phi,t\rangle$, and any constants?

   
   
   
   
   
   
   

8. The $n=2$, $l=1$ hydrogen atom radial wave functions is
   
   $$R_{21}(r) = N r e^{-Zr/2a_0}$$
   
   
   where $a_0=\hbar^2/(m_e e^2)$. What is the correct normalization factor $N$?

   (a)	$Z/\sqrt{3}a_0$	(c)	$(Z/a_0)^{5/2} (1/\sqrt{24})$	(e)	$(Z/2a_0)^3$
   (b)	$(Z/2a_0)^{3/2}$	(d)	$(Z/a_0)^5(1/24)$

9. For the rectangular barrier (the red, solid curve in the figure) we calculated the transfer matrix using ${\bf t} = {\bf d_{12} p_2 d_{21} p_1^{-1}}$ where the subscripts refer to the different regions in the plot. How would the matrices in ${\bf t}$ change for the barrier shown by the blue, dashed curve in the figure? See the last page of this exam for the forms of the discontinuity and propagation matrices.

   

10. What is the transmission probability due to the tunnel effect of a $1~eV$ electron incident on a barrier $0.5~nm$ wide and $5~eV$ high?

    (a)	$1.4\times 10^{-4}V$	(c)	$8.6\times 10^{-20}$	(e)	$3.2\times 10^{-30}$
    (b)	$3.6\times 10^{-14}$	(d)	$4.5\times 10^{-5}$

    
    

Problems. Put solutions on a separate piece of paper. Clearly show all work for full credit.

1. (11 pts.)

The $z$ component of the angular momentum operator is $$\hat{~L_z} = - i \hbar \left ( x {d \over d y} - y {d \over d x} \right ) \qquad .$$ We want to express this operator in terms of the spherical coordinates $r,\theta, \phi$. The transformation from Cartesian coordinates to spherical coordinates is the following. $$\begin{eqnarray*} x &=& r \sin \theta \cos \phi \\ y &=& r \sin \theta \sin \phi \\ z &=& r \cos \theta \end{eqnarray*}$$ One of the necessary steps is to show $${\partial r \over \partial x} = {x \over r} = \sin \theta \cos \phi$$ recalling that $r = \sqrt{x^2 + y^2 + z^2}$. Do that.

2. (11 pts)

At what speed is the DeBroglie wavelength of an $\alpha$ particle equal to that of a $\rm 5-keV$ photon?

3. (11 pts)	What is the expectation of momentum $\langle p \rangle$ for a particle in the following state? $$\psi(x,t) = Ae^{-(x/a)^2}e^{-i\omega t} \sin kx$$
4. (11 pts)	At time $t=0$, a hydrogen atom is in the superposition state $$\vert\psi(\vec r, 0)\rangle = \frac{1}{\sqrt{2}}\vert 100\ra... ... 4\sqrt{21}A\vert 21-1\rangle - i 4\sqrt 3 A \vert 321\rangle$$ where the eigenfunctions $\vert nlm\rangle$ are defined by the principle quantum number $n$, the angular momentum quantum number $l$, and the $z$-component of the angular momentum $m$ and $A=1/\sqrt{864}$. What is the probability a measurement of $L^2$ finds the value $\hbar^2 l (l+1)$?
5. (11 pts)	Recall our old friend the radioactive decay law $$\frac{dN}{dt} = - \lambda N$$ where $N$ is the number of radioactive nuclei and $\lambda$ is the decay constant. At $t=0$ the number of nuclei is $N_0$. The solution of this equation has been known to us at least since you took Intermediate Lab, but now solve this differential equation using the Method of Frobenius (i.e. the power series method), generate the recursion relationship, and make the appropriate choice of the leading coefficient to obtain that well known solution.
6. (15 pts)	For the one-dimensional step barrier shown in red in the figure below and given that $E<V_0$, what is the reflection coefficient in terms of $V_0$, $E$, $k_1$ and $k_2$? The wave numbers are $k_1 = \sqrt{2mE/\hbar^2}$ and $k_2 = \sqrt{2m(E-V_0)/\hbar^2}$. Make sure the parameters in your final answer are all real quantities.

Physics 309 Equations

$$E = h\nu = \hbar \omega \qquad v_{wave} = \lambda \nu \qquad... ...\vert^2 \qquad \lambda = {h \over p} \qquad p = \hbar k \qquad$$

$$-{\hbar^2 \over 2 m} {\partial^2 \over\partial x^2} \Psi(x,t... ...{A~}\rangle = \int_{-\infty}^{\infty} \psi^* \hat {A~} \psi dx$$

$$\langle\phi_{n'} \vert \phi_n \rangle = \int_{-\infty}^{\i... ... \delta(k - k') \quad e^{i\phi} = \cos\phi + i\sin\phi \quad$$

$$\vert\psi\rangle = \sum b_n \vert\phi_n\rangle \rightarrow ... ...angle dk \rightarrow b(k) = \langle\phi(k) \vert \psi \rangle$$

$$\vert\psi (t) \rangle = \sum b_n \vert\phi_n\rangle e^{-i\om... ...gle = \int b(k) \vert\phi(k)\rangle e^{-i\omega(k) t} dk \quad$$

$$\left [ \hat A,\hat B \right ] = \hat A \hat B - \hat B \hat... ... (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle^2 \qquad$$

The wave function, $\Psi(\vec r,t)$, contains all we know of a system and its square is the probability of finding the system in the region $\vec r$ to $\vec r + d\vec r$. The wave function and its derivative are (1) finite, (2) continuous, and (3) single-valued ( $\psi_1(a) = \psi_2(a)$ and $\psi^\prime_1(a) = \psi^\prime_2 (a)$) .

$$V_{HO} = {\kappa x^2 \over 2} \quad \omega = 2 \pi \nu = \sq... ...ad \vert\phi_n\rangle = A_ne^{-u^2/2}H_n(u) \quad u = \beta x$$

$$\beta^2 = {m\omega_0 \over \hbar} \quad \left ( \matrix{ \h... ...dagger \vert\phi_n\rangle = \sqrt{n+1} \vert\phi_{n+1} \rangle$$

$$\psi_1 = {\bf t} \psi_3 = {\bf d_{12} p_2 d_{21} p_1^{-1... ..._{x_0}^{x_1} \sqrt {2m(V(x) - E) \over \hbar^2} ~ dx\right ]$$

$$E = {\hbar^2 k^2 \over 2 m} \quad k = \sqrt{2m (E-V) \over \... ... {\rm incident\ flux}} \quad {\rm flux} = \vert\phi \vert^2 v$$

$$V(r) = {Z_1 Z_2 e^2 \over r} \quad \frac{dN}{dt} = {d\sigma ... ...\over 2} \right ) } \quad E = \frac{1}{2}\mu v^2 + V(r) \quad$$

$$\psi(x) = \sum_{n=1}^\infty a_n x^n \quad \langle K \rangle ... ... L = \vec r \times \vec p = I\vec \omega \quad I = \int r^2 dm$$

$$\vec p = m \vec v \quad KE_{rot} = \frac{L^2}{2I} \quad V = ... ...frac{L^2}{2\mu r^2} + V(r) \quad \mu = \frac{m_1 m_2}{m_1+m_2}$$

$$d\tau = r^2 d\cos\theta d\phi \quad L_z \vert nlm\rangle = m... ... L^2 \vert nlm\rangle = l (l+1) \hbar^2 \vert nlm\rangle \quad$$

Physics 309 Conversions, and Constants

Avogadro's Number ($N_A$)	$6.022\times 10^{23}$	fermi ($fm$)	$10^{-15}~m$
Boltzmann constant ($k_B$)	$1.381\times 10^{-23}~J/K$	angstrom ($\rm\AA$)	$10^{-10}~m$
	$8.62\times 10^{-5}~eV/k$	electron-volt ($eV$)	$1.6\times 10^{-19}~J$
Planck constant ($h$)	$6.621 \times 10^{-34}~J-s$	MeV	$10^6~eV$
	$4.1357\times 10^{-15}~eV-s$	GeV	$10^9~eV$
Planck constant ($\hbar$)	$1.0546\times 10^{-34}~J-s$	Electron charge ($e$)	$1.6\times 10^{-19}~C$
	$6.5821\times 10^{-16}~eV-s$	$e^2$	$\hbar c / 137$
Planck constant ($\hbar c$)	$197~MeV-fm$	Electron mass ($m_e$)	$9.11\times 10^{-31}~kg$
	$1970~eV-{\rm\AA}$		$0.511~MeV/c^2$
Proton mass ($m_p$)	$1.67\times 10^{-27}kg$	atomic mass unit ($u$)	$1.66\times 10^{-27}~kg$
	$938~MeV/c^2$		$931.5~MeV/c^2$
Neutron mass ($m_n$)	$1.68\times 10^{-27}~kg$	Speed of light ($c$)	$2.9979\times 10^8 ~m/s$
	$939~MeV/c^2$

Transfer Matrices

$${\bf d_{ij}} = \frac{1}{2} \left (\begin{array}{cc} 1+ \frac... ... e^{-ik_i2a} & 0 \\ 0 & e^{ik_i2a} \end{array} \right ) \quad$$