Calicut University 2007 M.Sc Physics PHY 102- Quantum Mechanics - Question Paper
FIRST SEMESTER M.Sc. DEGREE EXAMINATION, JANUARY 2007
Branch III- Physics
PHY 102- QUANTUM MECHANICS - 1
(2003 admission onwards)
(Pages: 2)
27977 Name....
Reg No.
FIRST SEMESTER M.Sc. DEGREE EXAMINATION, JANUARY 2007
Branch III - Physics PHY 102 - QUANTUM MECHANICS - I ( 2003 admission onwards)
Time: Three Hours Maximum : 80 Marks
Part A
Answer any five questions. Each question carries 4 marks.
/
1. What is a linear vector space? What is its relevance in quantum mechanics?
2. Justify the statement: The evolution in time of a state vector could be viewed as the continuous unfolding of a unitary transformation.
3. Explain the circumstances that led to a redefintion of angular momentum in quantum mechanics.
4. What are Paulis spin matrices? Show that an arbitrary 2x2 matrix can always be expressed as a linear combination of the Pauli matrices and the 2x2 identity matrix.
5. Bring out the meaning of the time reversal invariance. Show that the time reversed state corresponding to is (r, -t).
6. In a triplet state of a two-electron system, the spins are said to be parallel. Show that they are actually at an angle of about 70 to each other.
ii.. j'i.rimi lVavc method fails while dealing with high-energy scattering.
_ 8. State and explain optical theorem in scattering theory.
(5 x 4 = 20 marks.)
Part B
Answer both questions. Each question carries 20 marks.
9.(a) (i) Establish with sufficient justification that the probability amplitude in momentum space is the Fourier transform of its counterpart in co-ordinate space.
\/ (ii) Solve the problem of one-dimensional harmonic oscillator in the Heisenberg picture and obtain the matrices for H, x and p where the symbols have their usual meanings.
(7 13 = 20 marks.)
Or
(b) (i) Set up the Schrodinger equation for the hydrogen atom in spherical polar co-ordinates.
(ii) Separate the equation into radial and angular parts.
(iii) Solve the radial equation and obtain its eigen values and eigcn functions by assuming the eigen values of L2. y
2.7877
Show that t he wave function of a system of identical particles is either totally symmetric or totally antisymmetric.
(ii) How will you construct normalized wave functions for a system of (A) N bosons (B) N fcrmions?
(iii) Considering the helium atom as a two-eh;ctron system, show that the singlet state is always higher in energy than the triplet state.
10.(a) (i)
(4 + 7 + 9 = 20 marks.)
Or
Set up the integral equation of scattering and obtain an expression for the scattering amplitude in the Born approximation.
(ii) Illustrate the use of the above theory by taking the particular case of the central potential.
(12 + 8 = 20 marks.) (2 x 20 = 40 marks)
Part C
Answer any two questions. Each question carries 10 marks.
The Hamiltonian of a physical system with a three-dimensional state space is represented by the matrix
(b) (i)
11.
/ 2 1 OX H = I 1 2 0
What are the possible results when the energy of the system is measured?
(0
(ii)
12.
1 ( * \
If the system is in the state represented by =. J t I, find (H) , {H2) and the standard deviation,
AH.
One of the solutions of the Schrodinger equation for a simple harmonic oscillator of spring constant Jfc and mass rn is of the form ifi(x) = Ae~ax . Determine fully the wave function and the energy in this state. Given _
If the cigen functions of orbital angular momentum operators L2 and Lz with eigen values 1 = 1 and m = 1,0, 1 are denoted by calculate the result of operating on </>i,</>-i with
13.
14.
Lx. Hence find the eigen functions and the corresponding eigen values of Lx.
Calculate the partial wave cross-section , cto , for the case of S-wave scattering by a rigid sphere defined by the potential energy
v(r) _ / if i < a \ 0 if r > a
where a is the distance of closest approach.
(2 x 10 = 20 marks.)
9.
|
Attachment: |
| Earning: Approval pending. |
