Quantum Physics 2011/12
Tutorial Sheet 4: Time-dependence
An asterisk denotes a harder problem, which you are nevertheless encouraged to try!
1. An easy one to start with! A particle moving in the innite 1-d square well potential
V (x) = 0
for |x| < a,
V
Quantum Physics 2011/12
Solutions to Tutorial Sheet 4: Time-dependence
1. An easy one to start with! A particle moving in the innite 1-d square well potential
V (x) = 0
for |x| < a,
V (x) =
for |x| > a
is set up in the initial state (t = 0) described by
Quantum Physics 2011/12
Solutions to Tutorial Sheet 3: More Perturbations
1. Quarkonium is a system consisting of a heavy quark of mass mQ bound to its antiquark,
also of mass mQ . The inter-quark potential is of the form
a
V (r) = + br ,
r
where a, b are
Quantum Physics 2011/12
Tutorial Sheet 5: Variational Method
An asterisk denotes a harder problem, which you are nevertheless encouraged to try!
1. Estimate the ground-state energy of a 1-dimensional simple harmonic oscillator using
as trial function
(a)
Solutions to Tutorial Sheet 1: Mainly revision
1. Given the expansion of an arbitrary wavefunction or state vector as a linear superpo
sition of eigenstates of the operator A
(r, t) =
ci (t)ui (r) or |, t =
i
ci (t) |ui
i
use the orthonormality properties
Quantum Physics 2011/12
Solutions to Tutorial Sheet 5: Variational Method, Molecules
1. Estimate the ground-state energy of a 1-dimensional simple harmonic oscillator using
as trial function
(a) a (x) = cos x for |x| < /2, zero elsewhere,
(b) b (x) = 2 x2
Quantum Physics 2011/12
Tutorial Sheet 3: More Perturbations
An asterisk denotes a harder problem, which you are nevertheless encouraged to try!
1. Quarkonium is a system consisting of a heavy quark of mass mQ bound to its antiquark, also of
mass mQ . The
Quantum Physics 2011/12
Solutions to Tutorial Sheet 2: Perturbations
A note about notation
In perturbation theory we need distinguish between dierent unperturbed systems (hydrogen, square well, SHO) perturbed and unperturbed systems, order of perturbation
Quantum Physics 2011/12
Tutorial Sheet 2: Perturbations
An asterisk denotes a harder problem, which you are nevertheless encouraged to try!
The following trigonometric identities may prove useful in the rst two questions:
cos2 A
sin 2A
sin A sin B
cos A c
Quantum Physics 2011/12
Tutorial Sheet 1: Mainly revision
You will need to consult your notes from Junior Honours Quantum Mechanics and/or one
of the many textbooks on Quantum Mechanics.
1. Given the expansion of an arbitrary wavefunction or state vector