The operators a and a^{†} when acting on the energy eigenstates of
the harmonic oscillator, denoted by |n>, have the property

a^{†}|n> = (n + 1)^{½}|n + 1>, a|n> = n^{½}|n - 1>.

We have x = (a + a^{†})/(2α), p = -i(a - a^{†})/(2β), where α
=√(mω/(2ħ)), β =1/√(2mωħ).

Find the mean value and root mean square deviation of p^{2}, when the
oscillator is in the energy eigenstate |n>.

Through some coincidence, the Balmer lines from singly ionized helium in a distant star happen to overlap with the Balmer lines from hydrogen in the sun. Assuming that the relative motion is along the line of sight, how fast is that star receding from us?

Consider a simple, small, but macroscopic LC circuit made from conventional superconducting material and kept at a temperature below the critical temperature (~1 K). The circuit has no resistance.

(a) Write down a second order differential equation describing the time
evolution of the magnetic flux Φ = LI in the inductor.

(Consider this "the equation of motion" of the circuit.) Relabel C = m, k =
1/L, and compare this equation with the equation of motion of a simple harmonic
oscillator.

(b) Write down a Lagrangian for the LC circuit. (Lagrange's equation then is
the "equation of motion" of the circuit.) Find the generalized momentum
corresponding to the generalized coordinate in the Lagrangian, and write down
the Hamiltonian for the LC circuit.

(c) Assume that the circuit is cooled down to a
temperature of near zero K (~1 mK). At such a low temperature, only the
lowest allowed energy states are accessible to the system and observable
quantum-mechanical effects can appear. Quantize the system and find the
ground state energy of the system,

(d) Qualitatively reason why excited states of such a system would be unstable
even at very low temperature.

(a) Write down the normalized ground-state wave function for the hydrogen atom.

(b) Find <r> and Δr for the ground-state of the hydrogen atom.

(c) Find <1/r> and Δ(1/r) for the ground-state of the hydrogen atom.

∫_{0}^{∞} x^{n} e^{-ax} dx = n!/a^{n+1}

An exotic atom consists of a Helium nucleus (Z = 2) and an electron and an
antiproton p(bar) both in n = 2 states. Take the mass of the p(bar) to be 2000
electron masses and that of the helium nucleus to be 8000 m_{e}. For an
electron in the n = 1 state of hydrogen E = -13.6 eV.

(a) How much energy is required to remove the electron from this atom?

(b) How much energy is required to remove the p(bar) from this atom?

(c) Assume both the p(bar) and the electron are in 2p states. Then each can
de-excite to their ground state. It is observed that radiation always
accompanies those transitions when the electron jumps first, but when the p(bar)
jumps first there is often no photon emitted. Explain!