Section A
November 11, 2010
EP 341 Quantum Physics Exam1;
Time Interval: 17:0018:30
Name:
ID No:
Lecturer: Prof. Dr. Ramazan Koç
01 02
03
04
05
06
07 08 Total
Questions 1345 are 15 points and 2678 are 10 points.
Answer the following questions
1.
Write down time dependent and time independent
Schrödinger
equation, Heisenberg Uncertainty principle and de Broglie relation. Briefly explain how to use the equations.
Answer 1
(Time dependent Schrödinger equation. It is used to calculate energy levels and time
dependent wave function of the particles)
(Time independent Schrödinger equation. It is used to calculate energy levels and wave
function of the particles)
(Heisenberg Uncertainty relation, Means that position and momentum of a particle can not be measured
simultaneously.)
(Wavelength of a particle (matter wave) is inversely proportional to the momentum of the particle)
2.
Show that
is one of the solutions of the time
dependent Schrödinger
equation with zero potential.
Answer 2.
We substitute given wave function in to time dependent Schrödinger equation with V(x)=0:
Differentiation of the functions yields:
Then the terms
lhs=rhs
.
Barrier
Barrier
Well
Figure 1
Hints:
;
;
;
;
;
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November 11, 2010
3.
Evaluate the following commutators (
is momentum and
is position)
a)
;
b)
;
c)
What is the physical meaning of cummutators in quantum mechanics?
Answer 3
. Using definition of commutator of two operator;
and operator representation of
. We can write:
a)
then the commutator
b)
then the commutator
c)
If two operators can commute then they can be measured simultaneously, there is no uncertainty between
operators and they have the same eigenfunctions.
If two operator cannot commute then they cannot be measured simultaneously, there is an uncertainty between
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