Aligarh Muslim University (AMU) 2009 B.Sc Physics Quantum Mechanics - Question Paper
4290
2008-2009
B.Sc.(HONS.) (PART-III) EXAMINATION ( PHYSICS)
QUANTUM MECHANICS (PH-303)
Maximum Marks: 40 Duration : Three Hours.
All the questions arc compulsory.
The marks arc shown against the questions.
1.(a) Qualitatively discuss the difference between intrinsic and orbital angular momentum. Briefly discuss whether it would be corrcct to say that an angular momentum offi must be orbital and not intrinsic. 1+1
(b) Briefly discuss whether the product of two hermitian operators is also hermitian. 02
(c) Prove that
where the symbols have their usual meaning.
Briefly say what the equation tells us. 2+1
2.(a) Explain what is meant by a stationary quantal state? 01
(b) Briefly point out that the free particle is spread out all over the space. 02
(c) Calculate the particle flux associated with the free particle wave function Ae ,kx 02
(d) Normalize the one- dimensional wave function e "ax of a single particle, situated in the region 0 < x <+ oo. 02
3. Solve the one- dimensional time- independent Schrodinger equation for a onedimensional square well potential of a finite size and depth. Obtain the condition
for a bound state to occur. Tell how many levels are bound in the potential of
2 -h2
depth Vo and size a when Vo a = ~2m' , where m is the mass of the particle inside it. 4+2
OR
3. What is the Zero-Point Energy of the oscillator? Relate it to the Uncertainty
Principle. 24
4. Starting from the first principles, obtain the eigen functions and the energy cigcn values of the Rigid Rotator.
-2-
Where does the Rigid Rotator occur in physical problems? 5+1
OR
4(a) What is meant by the degeneracy of an energy level? 01
(b) Discuss the degeneracy of the levels of the Rigid Rotator. 02
(c) Obtain the number of the degenerate levels of the hydrogen- like atom for n =2.
Name an agency that can break the degeneracy of the levels of the hydrogen - like atom. 2+1
5.(a) Briefly tell what first led to the picture of the spinning electron. 02
(b) Write down all the relevant mutual commutators of the operators of the electron, spin components S x, S y and S z as also of a component, say S z and S . 03
(c) Briefly discuss whether the coordinate x commutes with S x. 02
OR
5(a) What is Zeeman Effect? Explain the terms Normal Zeeman Effect and Anomalous Zeeman Effect. 03
(b) With the help of the selection rules, discuss the transitions between two
hydrogen atom levels when the system is placed in either a weak or a strong magnetic field ( you may choose either the weak or the strong case). 04
6.(a) Define the terms : elastic scattering cross section, scattering amplitude and phase
shift. 03
(b) If f ( 0 ,<(>) is the unsymmetrized scattering amplitude; write down the cross section, for fermions, when the incident particle is identical with the target. Give one practical example. 3+ 1
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