Homework 4

  1. The 1D model of a crystal puts the following constraint on the wave number $k$.

    \begin{displaymath}\cos(\phi)=\cos(ka)+{ma^2V_0\over\hbar^2}{\sin(ka)\over ka} \end{displaymath}

    Assume that ${ma^2V_0\over\hbar^2}={3\pi\over 2}$ and plot the constraint as a function of $ka$. Plot the allowed energy bands on an energy axis assuming $V_0=2$ eV and the spacing between atoms is 5 Angstroms.

  2. In a 1D square well, there is always at least one bound state. Assume the width of the square well is $a$. By the uncertainty principle, the kinetic energy of an electron localized to that width is ${\hbar^2\over 2ma^2}$. How can there be a bound state even for small values of $V_0$?

  3. At $t=0$ a particle is in the one dimensional Harmonic Oscillator state $\psi(t=0)={1\over\sqrt{2}}(u_0+u_1)$. Is $\psi$ correctly normalized? Compute the expected value of $x$ as a function of time by doing the integrals in the $x$ representation.

  4. Prove the Schwartz inequality $\left\vert\langle\psi\vert\phi\rangle\right\vert^2\leq\langle\psi\vert\psi\rangle\langle\phi\vert\phi\rangle$. (Start from the fact that $\langle\psi+C\phi\vert\psi+C\phi\rangle\geq 0$ for any $C$.

  5. The hyper-parity operator $H$ has the property that $H^4\psi=\psi$ for any state $\psi$. Find the eigenvalues of $H$ for the case that it is not Hermitian and the case that it is Hermitian.

  6. Find the correctly normalized energy eigenfunction $u_5(x)$ for the 1D harmonic oscillator.

  7. A beam of particles of energy $E>0$ coming from $-\infty$ is incident upon a double delta function potential in one dimension. That is $V(x)=\lambda\delta(x+a)-\lambda\delta(x-a)$.
    Find the solution to the Schrödinger equation for this problem.
    Determine the coefficients needed to satisfy the boundary conditions.
    Calculate the probability for a particle in the beam to be reflected by the potential and the probability to be transmitted.

Jim Branson 2013-04-22