Physics 309 Final, Fall 2014

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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. The hydrogen atom eigenfunctions can be represented as ${ \vert nlm \rangle}$. Describe the meaning of each quantum number.

   
   
   
   
   
   
   
   

2. Why did we use the center-of-mass coordinate system when we studied the $\rm CO$ molecule?

   
   
   
   
   
   
   
   

3. What is the quantum program?

   
   
   
   
   
   
   
   
   

4. Recall the absorption spectrum of carbon monoxide shown in the figure. How would the spectrum change if the $\rm CO$ molecule did not rotate. Explain.

   

   
   
   
   

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5. What is the definition of a solid angle?

   
   
   
   
   
   
   
   

6. In solving the Schroedinger equation for the harmonic oscillator potential we rewrote the Schroedinger equation in the form
   
   $$\frac{d^2\phi}{du^2} + \left (\frac{\alpha}{\beta^2} - u^2 \right) \phi = 0$$
   
   
   where $u = \beta x$, $x$ is the position of the particle, and $\beta$ and $\alpha$ are constants. What is the asymptotic form of this differential equation? Explain your reasoning.

   
   
   
   
   
   
   
   

7. A hydrogen atom is in the following initial state.
   
   $$\vert \psi(r,\theta,\phi,t=0) \rangle = A\left (\vert 321\rangle + 3\vert 211\rangle\right )$$
   
   
   What is the value of the normalization constant? Show your reasoning for full credit.

   
   
   
   
   
   
   
   

8. The figure below shows the energy diagram we used to calculate alpha decay. What would happen to the calculation of the lifetime if the energy $E_\alpha$ is increased? Explain.

   

   
   
   

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9. A particle in a box of width $a$ is in the initial state
   
   $$\vert\psi\rangle = \frac{3\vert\phi_2\rangle + 7\vert\phi_7\rangle}{\sqrt{58}}$$
   
   
   where $\vert\phi_n\rangle$ is the eigenfunction of the $n^{th}$ energy level. The position of the particle is measured and a value $x_p$ is found. What would subsequent a subsequent measurement of the energy of the particle find? Explain.

   
   
   
   
   
   
   
   

10. A hydrogen atom is in its third excited state. What is the principle quantum number $n$ of the state it must jump to in order to absorb light with the longest possible wavelength? Explain.

    
    
    
    
    
    
    
    

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. $$\begin{eqnarray*} x &=& r \sin \theta \cos \phi \\ y &=& r \sin \theta \sin \phi \\ z &=& r \cos \theta \end{eqnarray*}$$ Show $${\partial \phi \over \partial x} = -{\sin \phi \over r \sin \theta} \qquad .$$ Hint: Use ${\partial (\tan \phi) / \partial x}$ where $\tan \phi = y/x$.

2. (11 pts)	With $C_0=1$ in the recurrence relationship for the hydrogen atom $$C_{i+1} = \frac{(i+l+1) - \lambda}{(i+1)(i+2l+2)}C_i$$ obtain $C_1$ and use $$\begin{eqnarray*} u_{nl}(\rho) = A_{nl} e^{-\rho/2} \rho^{l+1} \sum_{i=0}^{n-l-1... ...i\rho^i \\ \rho = 2\kappa_n r \qquad \kappa_n = \frac{1}{a_0 n} \end{eqnarray*}$$ to show $$u_{20}(r) = \frac{A_{20}}{a_0} r e^{-r/2a_0} \left( 1 - \frac{r}{2 a_0} \right )$$ where $a_0$ is the Bohr radius. You do NOT have to calculate $A_{20}$.
3. (11 pts)	The general solution to the classical harmonic oscillator is $x(t)=A\sin(\omega_0 t + \delta)$. Starting from this equation get an expression for the period of the motion (the time to make one complete oscillation) in terms of the parameters of the general solution. How is this result related to the frequency?
4. (11 pts)	For the state $$\psi(x,t) = \frac{1}{\sqrt{a\sqrt{2\pi}}}\exp{\left [-\frac{... ...]}\exp{\left (\frac{ip_ox}{\hbar}\right )}\exp{(-i\omega_0 t)}$$ show that $$(\Delta x )^2 = a^2$$ If the parameter $a$ changes, then what happens to $\vert\psi^2\vert$?
5. (13 pts)	Consider a case of one dimensional nuclear `fusion'. A neutron is in the potential well of a nucleus that we will approximate with an infinite square well with walls at $x=0$ and $x=a$. The eigenfunctions and eigenvalues are $$\begin{eqnarray*} E_n = {n^2 \hbar^2 \pi^2 \over 2 m a^2} \qquad \phi_n & = & \... ... a \\ & = & 0 \hspace{3.5cm} x < 0 ~ {\rm and} ~ x > a \qquad. \end{eqnarray*}$$ The neutron is in the $n=9$ state when it fuses with another nucleus that is twice its size, instantly putting the neutron in a new infinite square well with walls at $x=0$ and $x=3a$. See next page.

5. (cont.)

What are the new eigenfunctions and eigenvalues of the fused system? Which state, if any, in the fused system will have the same energy as the original state? Calculate the probabilities for finding the neutron in the two lowest energy states of the fused system.

6. (13 pts)

The general solution to the rectangular barrier problem for the potential shown in the figure is $$\begin{eqnarray*} \psi_1 &=& Ae^{ik_1 x} + Be^{-ik_1 x} \newline \psi_2 &=& Ce^{... ...} + De^{-ik_2 x} \newline \psi_3 &=& Fe^{ik_1 x} + Ge^{-ik_1 x} \end{eqnarray*}$$ where the wave numbers are defined as follows. $$k_1 = \sqrt{ 2mE \over \hbar^2 } \qquad k_2 = \sqrt{ 2m(E-V_0) \over \hbar^2}$$ We expressed the wave functions in each region in the form of column vectors $$\psi_1 = \pmatrix{ A \cr B } \qquad \psi_2 = \pmatrix{ C \cr D } \qquad \psi_3 = \pmatrix{ F \cr G }$$ and the boundary conditions in the form of the matrices $$\psi_1 = {\rm\bf d}_{12} {\rm\bf p}_2 {\rm\bf d}_{21} {\rm\bf p}_{1}^{-1}\psi_3 = {\rm\bf t}\psi_3$$ (1) where ${\rm\bf t}$ is the transfer matrix, ${\rm\bf d_{12}}$ and ${\rm\bf d_{21}}$ are discontinuity matrices, and ${\rm\bf p_2}$ and ${\rm\bf p_1^{\bf -1}}$ are the propagation matrices. The discontinuity and propagation matrices are defined on the equation sheet. Consider the elements of $\rm\bf d_{12}$, $\rm\bf d_{21}$, ${\rm\bf p_{2}}$, and ${\rm\bf p_{1}^{-1}}$ to be known quantities. In region 3 we set $G=0$ because no waves were incident from the right. The coefficient $A$ represents the incident wave coming from the left. It is our `beam' and so we consider it to be known. It turns out we can neglect the effect of the final propagation matrix ${\bf p_1^{-1}}$ in Equation 1 for the calculation of transmission and reflection coefficients. The revised expression is the following. $$\psi_1 = {\rm\bf d}_{12} {\rm\bf p}_2 {\rm\bf d}_{21} \psi_3 = {\rm\bf t}\psi_3$$ (2) Starting from Equation 2 above derive a relationship for the coefficient $F$ in terms of $A$ and the wave numbers $k_1$ and $k_2$. Hint: Be judicious in your efforts. You don't have to calculate every element of the transfer matrix.

Physics 309 Equations

$$E = h\nu = \hbar \omega \qquad v_{wave} = \lambda \nu \qquad... ...mbda = {h \over p} \qquad p = \hbar k \qquad \vec p = m \vec v$$

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

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

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

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

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

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 $\psi^\prime_1(a) = \psi^\prime_2 (a)$) .

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

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

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

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

$$V(r) = {Z_1 Z_2 e^2 \over r} \quad \frac{dN}{dt} = {d\sigma ... ...\frac{1}{2}\mu v^2 + V(r) \quad \mu = \frac{m_1 m_2}{m_1+m_2}$$

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

$$KE_{rot} = \frac{L^2}{2I} \quad E_l = \frac{l(l+1)\hbar^2}{2... ...ad ME = \frac{p_r^2}{2\mu} + \frac{L^2}{2\mu r^2} + V(r) \quad$$

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

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

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