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  1. Monochromatic light with a wavelength of $6000 \AA$ passes through a fast shutter that opens for $10^{-9}$ sec. What is the subsequent spread in wavelengths of the no longer monochromatic light?

  2. Calculate $\langle x\rangle$, $\langle x^2\rangle$, and $\sigma_x$, as well as $\langle p\rangle$, $\langle p^2\rangle$, and $\sigma_p$, for the normalized wavefunction

\psi(x) = \sqrt{\frac{2 a^3}{\pi}} \frac{1}{x^2+a^2}.

    Use these to find $\sigma_x \sigma_p$. Note that $\int_{-\infty}^{\infty} dx/(x^2+a^2) = \pi/a$.

  3. Classically, if a particle is not observed then the probability of finding it in a one-dimensional box of length $L$, which extends from $x=0$ to $x=L$, is a constant $1/L$ per unit length. Show that the classical expectation value of $x$ is $L/2$, the expectation value of $x^2$ is $L^2/3$, and the standard deviation of $x$ is $L/\sqrt{12}$.

  4. Demonstrate that if a particle in a one-dimensional stationary state is bound then the expectation value of its momentum must be zero.

  5. Suppose that $V(x)$ is complex. Obtain an expression for $\partial P(x,t)/\partial t$ and $d/dt \int P(x,t) dx$ from Schrödinger's equation. What does this tell us about a complex $V(x)$?

  6. $\psi_1(x)$ and $\psi_2(x)$ are normalized eigenfunctions corresponding to the same eigenvalue. If

\int_{-\infty}^\infty \psi_1^\ast \psi_2 dx = c,

    where $c$ is real, find normalized linear combinations of $\psi_1$ and $\psi_2$ which are orthogonal to (a) $\psi_1$, (b) $\psi_1+\psi_2$.

  7. Demonstrate that $p=-{\rm i} \hbar \partial/\partial x$ is an Hermitian operator. Find the Hermitian conjugate of $a = x + {\rm i} p$.

  8. An operator $A$, corresponding to a physical quantity $\alpha $, has two normalized eigenfunctions $\psi_1(x)$ and $\psi_2(x)$, with eigenvalues $a_1$ and $a_2$. An operator $B$, corresponding to another physical quantity $\beta$, has normalized eigenfunctions $\phi_1(x)$ and $\phi_2(x)$, with eigenvalues $b_1$ and $b_2$. The eigenfunctions are related via
    $\displaystyle \psi_1$ $\textstyle =$ $\displaystyle (2 \phi_1+3 \phi_2) \left/ \sqrt{13},\right.$  
    $\displaystyle \psi_2$ $\textstyle =$ $\displaystyle (3 \phi_1-2 \phi_2) \left/ \sqrt{13}.\right.$  

    $\alpha $ is measured and the value $a_1$ is obtained. If $\beta$ is then measured and then $\alpha $ again, show that the probability of obtaining $a_1$ a second time is $97/169$.

  9. Demonstrate that an operator which commutes with the Hamiltonian, and contains no explicit time dependence, has an expectation value which is constant in time.

  10. For a certain system, the operator corresponding to the physical quantity $A$ does not commute with the Hamiltonian. It has eigenvalues $a_1$ and $a_2$, corresponding to properly normalized eigenfunctions
    $\displaystyle \phi_1$ $\textstyle =$ $\displaystyle (u_1+u_2)\left/\sqrt{2},\right.$  
    $\displaystyle \phi_2$ $\textstyle =$ $\displaystyle (u_1-u_2)\left/\sqrt{2},\right.$  

    where $u_1$ and $u_2$ are properly normalized eigenfunctions of the Hamiltonian with eigenvalues $E_1$ and $E_2$. If the system is in the state $\psi=\phi_1$ at time $t=0$, show that the expectation value of $A$ at time $t$ is

\langle A\rangle = \left(\frac{a_1+a_2}{2}\right) + \left(\f...
...a_1-a_2}{2}\right)\cos\left(\frac{[E_1-E_2] t}{\hbar}\right).

next up previous
Next: One-Dimensional Potentials Up: Stationary States Previous: Stationary States
Richard Fitzpatrick 2010-07-20