PX3509 (2007/08) - TUTORIAL SHEET 6
1. † The eigenfunctions of a hydrogen-like atom have the form ψnlm (x) = Rnl (r)Ylm (θ, ϕ).
1
(a) Explain in details the meaning of the quantum numbers n, l, m and write down all the relevant eigenvalue equations that the ψnlm satisfy. In your answer explain carefully the range of variability of each quantum number. (b) The states ψnlm comprise a orthonormal base. Write down a compact expression for the result of the scalar products (ψnlm , ψn l m ) (c) Be ψ(x) an arbitrary wave function describing the state of a hydrogen-like atom. How can you write its expansion in the basis {ψnlm }nlm ? (Hint: Since you have three quantum numbers labeling the element of the base, the expansion will contain three embedded summations. Have care of writing the precise range of variability of n, l and m) 2. The ground state of the hydrogen atom is described by ψ100 (x) = 1 πa3 0
1 2
exp −
r a0
,
where a0 = 0.53 × 1010 m is the Bohr radius and r is the radial distance from the nucleus. This state is spherical symmetric, as it does not depend upon the angles. (a) Checking the lecture notes about angular momentum, write down explicitly the form of the angular part Ylm (θ, ϕ) for the ground state. What is the angular momentum L of this state? (b) The probability to find the electron at a distance r from the nucleus within the radial element dr is P (r)dr = r2 |Rnl |2 dr, where P (r) taken alone is the radial probability density. (i) Using the result from part (a) find the expression for Rnl (r) in the ground state. (ii) Calculate for which r the radial probability density P (r) is maximum. (c) Calculate the expectation value of the radial distance for the electron in the hydrogen atom, i.e. evaluate the scalar product r = (ψ100 , rψ100 ). Finally compare the value of r with the result in part (b)-(ii). Can you explain qualitatively why the two results don’t coincide? (Hint: The scalar product in part (c) involves a 3-d space integration. The volume element in spherical coordinates is d3 x = drr2 dΩ,
PX3509 (2007/08) - TUTORIAL SHEET 6
2
where the angular part is dΩ = dθ sin θdϕ. Since the ground state ψ100 is spherically symmetric the angular integration can be carried out on its own and the result is dΩ = 4π, which is nothing else than the value of the area of the surface of the sphere of radius 1. You will need the integral
∞
dze−z z 3 = 6.
0
To understand why the most probable radius and the mean radius r are different, it can be useful to draw a qualitative radial plot of r2 |Rnl |2 .) 3. Consider again the hydrogen atom ground state as in problem 2. (a) Calculate the probability P (0 ≤ r ≤ 3a0 ) that the electron is within the radial range [0, 3a0 ]. (b) Without performing any further integration write down the value of the probability that the electron is located in the radial range [3a0 , ∞]. (c) Imagine to be a researcher, studying for the first time the hydrogen atom from a theoretical point of view. Pondering upon the two above results, how would you decide to define the concept of size for the hydrogen atom? (Hint: When calculating the integral pass to the adimensional variable z =
3 0 r a0 .
You will need
dzz 2 e−2z = 0.23 )
4.
An electron in a hydrogen atom is described by the superposition state 1 1 ψ = c300 ψ300 + ψ322 + ψ32−2 , 2 3 with c300 > 0. (a) Find the value of c300 that normalizes the state. (b) Is ψ an eigenstate of the energy? What is the expectation value E for the energy? ˆ (c) Is ψ an eigenstate of L2 ? Calculate the expectation values of L2 and Lz .
5.
Consider a harmonic oscillator perturbed by a quartic term αx4 , where α > 0. ˆ ˆ (a) Write down the unperturbed Hamiltonian H0 and the full, perturbed Hamiltonian H. (b) The unperturbed ground state is u0 (x) = 1 l0 π 1/2 e−x
2 /2l2 0
,
PX3509 (2007/08) - TUTORIAL SHEET 6
with the length scale l0 defined as
2 l0 ≡
3
µω
and where µ is the reduced mass and ω the frequency of the oscillator. Calculate the first order shift E0 of the unperturbed ground state energy E0 = ω/2. Hint:
+∞ 4 −z 2 −∞ z e (1) (0)
dz =
√ 3 π 4 ,
Answer: E0 =
(1)
3 2α 4µ2 ω 2
4 = 3 l0 α. 4
(c) The unperturbed first excited state is described by u1 (x) =
(1)
2x −x2 /2l0 2 e . 2 l0 π 1/2 l0
(0)
1
Calculate the first order shift E1 of the unperturbed energy E1 = 3 ω/2. Hint:
+∞ 6 −z 2 −∞ z e
dz =
√ 15 π 8 ,
Answer: E1 =
(1)
15 2 α 4µ2 ω 2
=
15 4 4 l0 α.
4 (d) Note that the energy levels corrections are small if l0 α use of first order perturbation theory is justified.
1. When this condition is satisfied the
In a transition from the first unperturbed excited state to the unperturbed ground state, the harmonic (0) oscillator emits one photon of frequency ω1→0 satisfying ω1→0 = E1 − E0 . (i) Find the value of ω1→0 . (ii) Now take into account the shift in the energy levels due to the perturbation and find a formula for the (1) modified transition frequency ω1→0 . Answer: ω1→0 = ω1→0 +
(1) (0)
4 3l0 α
(0)
(0)
(0)
(0)
.