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Exercise 3

1. The document provides examples of problems involving time-dependent perturbation theory. The first problem asks to calculate the time interval needed for a harmonic oscillator initially in the ground state to have the greatest probability of being found in the first excited state after a small constant force is applied for a time. 2. The second problem involves calculating the probability that the spin of a spin 1/2 system points in the positive z-axis under the influence of static and rotating magnetic fields. It also asks to discuss when perturbation theory breaks down and provide an exact solution to the time-dependent Schrodinger equation. 3. The remaining problems involve calculating Zeeman shifts of hydrogen energy levels, identifying electric dipole transitions in

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237 views2 pages

Exercise 3

1. The document provides examples of problems involving time-dependent perturbation theory. The first problem asks to calculate the time interval needed for a harmonic oscillator initially in the ground state to have the greatest probability of being found in the first excited state after a small constant force is applied for a time. 2. The second problem involves calculating the probability that the spin of a spin 1/2 system points in the positive z-axis under the influence of static and rotating magnetic fields. It also asks to discuss when perturbation theory breaks down and provide an exact solution to the time-dependent Schrodinger equation. 3. The remaining problems involve calculating Zeeman shifts of hydrogen energy levels, identifying electric dipole transitions in

Uploaded by

Max Jenkinson
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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PHYS 40202/6402 Advanced QM

Examples 3
(Starred questions are MSc coursework)

These exercises cover the section of the course on time-dependent perturbation


theory.
1. * A one-dimensional harmonic oscillator of mass m and angular frequency ω
in its ground state is subject to a small constant force F acting for a time
interval τ . What value of τ gives the greatest chance that the oscillator will be
found in its first excited state thereafter? The ground and first excited state
wavefunctions of the oscillator are
s
mωx2
1/4 !
mω 2mω

φ0 (x) = exp − , φ1 (x) = xφ0 (x)
πh̄ 2h̄ h̄
respectively.
2. * (a) The time-dependent Hamiltonian Ĥ = Ĥ0 + V̂ (t) with
eB0 eB1  
Ĥ0 = Ŝz , V̂ (t) = Ŝx cos ωt + Ŝy sin ωt
m m
describes a spin S = 1/2 system which is subject to the static magnetic field
(0, 0, B0 ) and the rotating magnetic field (B1 cos ωt, B1 sin ωt, 0). B0 and B1
are uniform throughout space. Initially, at time t = 0, the spin of the system
is pointing in the direction of the negative z-axis. Assuming B1 is weak and
taking V̂ (t) as perturbation, calculate the probability that at time t > 0 the
spin points in the positive z-axis.
(b) Discuss at what value of ω (i.e. resonance), the perturbation theory breaks
down for sufficiently large t, however weak B1 . Suggest a criterion in t for the
perturbation theory to hold.
(c) Solve exactly the time-dependent Schrödinger equation at the resonance for
the given initial condition. Discuss the relation between the exact results and
the first-order perturbation results obtained earlier.
3. A hydrogen atom is placed in a uniform magnetic field B in z direction. The
corresponding weak-field Zeeman energy shift is given by
∆E = gµB BMJ
where the Landé g factor is defined by
J(J + 1) + S(S + 1) − L(L + 1)
g =1+ ,
2J(J + 1)

1
with usual notations for angular momentum quantum numbers MJ , J, L, and
S. List the weak-field Zeeman shifts for all the n = 1 and n = 2 states of
hydrogen and draw a level diagram. Mark all the transitions consistent with
electric dipole selection rules.

4. * Which of the following transitions in carbon are electric dipole transitions?

(a) (2s)2 (2p)(3d) 3 D → (2s)2 (2p)2 3 P


(b) (2s)2 (2p)(3s) 3 P → (2s)2 (2p)2 1 S
(c) (2s)2 (2p)(3d) 1 D → (2s)2 (2p)(3s) 1 P
(d) (2s)(2p)3 3 D → (2s)2 (2p)2 3 P
(e) (2s)2 (2p)(3p) 3 P → (2s)2 (2p)2 3 P
(f) (2s)2 (2p)(3d) 1 D → (2s)2 (2p)2 1 S

5. The K1 and K2 mesons have slightly different masses m1 and m2 and lifetimes
τ1 = 0.9 × 10−10 s and τ2 = 0.5 × 10−7 s respectively. The K 0 and K̄ 0 mesons
ar the superpositions
1 1
|K 0 i = √ (|K1 i + |K2 i), |K̄ 0 i = √ (|K1 i − |K2 i).
2 2

A K 0 meson is produced in the process π − + p → Λ0 + K 0 at time t = 0. Show


that the probability that at a time t the stationary (relativistic) K 0 meson has
turned into a K̄ 0 meson is
(m1 − m2 )c2 t
!
1 −t/τ1 1 1 t
   
P = e + e−t/τ2 − 2 exp − + cos .
4 τ1 τ2 2 h̄

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