Applied Quantum Mechanics 2nd. Edition by Levi, A. F. J. | 9780521183994 | 0521183995 | Holooly

Quantum Mechanics

Applied Quantum Mechanics

89 SOLVED PROBLEMS

Question: 11.2

A hydrogen molecule (H2) consists of two identical hydrogen atoms separated by a center-to-center distance of approximately 2a = 01 nm and having a total moment of inertia I. (a) Show that the solution to the Schrödinger equation gives wave function Φm=Ae^imϕ where ϕ is the angle of rotation and A ...

Verified Answer:

(a)

\hat{H} \psi_{lm}=\frac{L^{2} }{2I}\psi...

Question: 11.3

Find pairs of two hydrogen atom principle quantum numbers n for which the difference in energy eigenvalue is the same and hence they give rise to coincident spectral lines. ...

Verified Answer:

We wish to find pairs of quantum numbers

(n...

Question: 11.4

Use first-order perturbation theory to estimate the energy shifts of the hydrogen 2s and 2p states due to the fact that the proton is not a point charge, treating it (for simplicity) as a uniformly charged hollow spherical shell of radius b = 5×10^-14 cm. Comment on these results. Compare the ...

Verified Answer:

The wave functions for the hydrogen atom are of th...

Question: 11.5

(a) What is the effect of applying a uniform electric field on the energy spectrum of an atom? (b) If spin effects are neglected, the four states of the hydrogen atom with quantum number n = 2 have the same energy, E^0. Show that when a uniform z-directed electric field E is applied to hydrogen ...

Verified Answer:

(a) We are asked to predict what the effect of app...

Question: 10.4

A particle of mass m and charge e confined to motion in the x direction oscillates in a one-dimensional harmonic potential with angular frequency ω. (a) Show, using perturbation theory, that the effect of an applied uniform electric field E in the x direction is to lower all the energy levels by ...

Verified Answer:

(a) The solution follows that already given in thi...

Question: 10.5

The potential seen by an electron with effective mass me^* in a GaAs quantum well is approximated by a one-dimensional rectangular potential well of width 2L in such a way that V(x) = 0 for 0 < x < 2L and V(x) = ∞ elsewhere. (a) Find the eigenvalues En, eigenfunctions ψn ,and parity of ψn. (b) The ...

Verified Answer:

(a) It is given that the potential seen by an elec...

Question: 10.6

In this chapter we solved for the first excited state of a two-dimensional harmonic oscillator subject to perturbation W = κ′xy. How do the three-fold degenerate energy E = 3ℏω and the four-fold degenerate energy E = 4ℏω separate due to the same perturbation? ...

Verified Answer:

A two-dimensional harmonic oscillator with motion ...

Question: 11.1

(a) Use the transformation between Cartesian and spherical coordinates x= r sin (θ)cos (ϕ), y= r sin (θ)cos (ϕ), z= r cos(θ), to obtain the expression Lz = -iℏ∂/∂ϕ. (b) Calculate [ϕ,Lz]. (c) Show that L² = -ℏ²(1/sin(θ)∂/∂θ(sin(θ)∂/∂θ)+1/sin²(θ)∂/∂ϕ²). (d) Derive [Ly,Lz] = iℏLx using ...

Verified Answer:

(a) For a particle at position

r

wi...

Question: 10.1

A particle of mass m moves in a one-dimensional, infinitely deep potential well having a parabolic bottom,V(x) = ∞ for ∣x∣ ≥ L and V(x) = ξx²/L² for -L<x<L where ξ is small compare with the ground-state energy. Treat the term ξ as a perturbation on the square potential well (denoting the ...

Verified Answer:

The eigenfunctions for a rectangular potential wel...

Question: 10.2

Calculate the energy levels of an anharmonic oscillator with potential of the form V(x) = κ/2 x²+ξx³ℏω where κ is the spring constant for a harmonic potential. Show that the difference between two adjacent perturbed levels is En-En-1 = ℏω(1-15ξ²(ℏ/mω)³n/2). A heterodiatomic molecule can absorb or ...

Verified Answer:

The Hamiltonian for the one-dimensional harmonic o...

Loading...