The wave function of a system in the ground state is given as

ψ(**r**,t) = [exp(-iωt)/(πa_{0}^{3})^{½}]
exp(-r/a_{0}).

(a) Sketch the probability density in coordinate space as
function of r/a_{0.
}(b) Find the momentum space wave function Φ(**p**,t).

Hint: Use spherical coordinates for evaluation of the
integral transform.

(c) Find the probability density function in momentum
space. Sketch it as a function of pa_{0}/ħ.

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!

The potential energy of the nuclei of a diatomic molecule as a function of
their separation r is given by

U(r) = -2D[a_{0}/r - a_{0}^{2}/r^{2}].

Here D is a constant with units of energy.

Approximate this potential energy function near its minimum by a harmonic
oscillator potential energy function and determine the vibrational energy levels
of the molecule with zero angular momentum.

The Rydberg constant, R_{H} = 109737.568525/cm is one of the most
accurately known fundamental constants.

(a) Find the wave number of the Balmer alpha line (n = 3 to n' = 2) in atomic
hydrogen. Neglect fine structure.

(b) Is the Balmer alpha line in atomic deuterium shifted towards the blue or
towards the red compared to normal hydrogen?

(c) Calculate the shift in wave number between deuterium and hydrogen.

An electron in the hydrogen atom occupies the combined position and spin state

R_{21}(r)[√⅓ Y_{10}(θ,φ)χ_{+} + √⅔ Y_{11}(θ,φ)χ_{-}].

(a) If you measure L^{2}, what value(s) might you get, and with what
probability(ies)?

(b) If you measure L_{z}, what value(s) might you get, and with what
probability(ies)?

(c) If you measure S^{2}, what value(s) might you get, and with what
probability(ies)?

(d) If you measure S_{z}, what value(s) might you get, and with what
probability(ies)?

(e) If you measured the position of the electron, what is the probability
density for finding the electron at r, θ, φ in terms of the variables given
above.

(f) If you measured both S_{z} and the distance of the electron from
the proton, what is the probability per unit length for finding the particle
with spin up a distance r from the proton in terms of the variables given above?

Useful integral: ∫_{0}^{π}sinθ dθ∫_{0}^{2π}dφ
|Y_{lm}(θ,φ))|^{2} = 1.