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Quantum Mechaics 1 Question Paper

Quantum Mechaics 1

Course:Bachelor Of Science Physics

Institution: University Of Nairobi question papers

Exam Year:2011

UNIVERSITY OF NAIROBI
FIRST SEMESTER EXAMINATION 2010/2011
3RD YEAR EXAM FOR DEGREE OF BACHELOR OF SCIENCE
SPH 309:QUANTUM MECHANICS 1

Q1. (a) Define Hermitian operators. (2marks)

(b) Prove that Eigenvalues of a Hermitian operators are real and eigenfunctions of distinct eigenvalues are orthogonal. (12marks)

(c) Prove that
u(x)=ce^ikx are the eigenfunctions of the momentum operator
p= -ih d/dx

Hence find the corresponding eigenvalue (6marks)

Q2. (a) For a particle in a box,the wave function is:
*(x,0) = C*2/(L/X) (sin 3.214x/(l/x)+ /2 sin 2*x)
Determine the normalization constant C. (5marks)

Q3. from the time-dependent Schrodinger equation,where H = H0 + Hg',H is assumed to be a simple time-independent Hamiltonian whose eigenvalue problem can be solved by HUk = EkUk,
gH' is assumed to be small but time dependent perturbation.
We expand * in terms of Uk.Since phi is time dependent,we use the eigen functions of the unperturbed time-dependent Schrodinger equation.
Using an expansion of the equation,cast the time-dependent Schrodinger equation.

(b) Solve the equation in (a) to obtain an expression for a(t) upto 1st order of perturbation.

Q4. Prove that the Clebsch-Gordon coefficient
<j1,m1,i1,/j1,j2,jm> = 0 unless m=m1+m2 (10marks)

(b) Prove that the Chlebs-Gordon coefficients are zero unless j1,j2 and j3 satify the
triangular condition <j1-j2> is less or equal to j1+j2. (10marks).

Quantum Mechaics 1 Question Paper

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