# 2 - TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK F ¨...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK F ¨ ORDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik f¨ ordjupningskurs SI2380 f¨or F4 Thursday December 20, 2007, 8.00 – 13.00 Write on each page: Name, study program and year, problem number Motivate in detail! Insufficient motivation leads to reduction of points Allowed material: Formula collection (handed out with the exam), BETA, pocket calculator Grading system: Max 3 points per problem Examiner: Jack Lidmar, tel 5537 8715 1 Variational calculation of a 1D bound state A particle moves in a potential V ( x ) = ( ∞ if x < ax if x ≥ where a > . Suggest a variational ansatz for the ground state wave function and calculate an estimate of its energy. Calculate also an estimate of the expectation value of the position. 2 Spin in a magnetic field Consider a spin S in a uniform magnetic field B , with a Hamiltonian ˆ H =- γ B · ˆ S . Derive the equation d dt D ˆ S E =- γ B × D ˆ S E . Specialize now to a spin- 1 2 particle and assume that the magnetic field is aligned in the z-direction, B = B e z . Assume further that the state at t = 0 is | ψ (0) i = 1 √ 2 ( | + i + |-i ) , where |±i are the eigenstates of ˆ S z with eigenvalues ± ¯ h/ 2, respectively. What is the state vector at time t = 2 π/γ | B | ? What is the expectation value D ˆ S x ( t ) E at the same time? You may need the commutation relations for the components of the spin: [ ˆ S x , ˆ S y ] = i ¯ h ˆ S z , and cyclic permutations of x,y,z. SEE NEXT PAGE! 1 3 Identical particles Two identical non-interacting spin- 1 2 particles are placed in a 1D potential V ( x ) = ( if 0 ≤ x ≤ L ∞ otherwise A strong magnetic field is applied so that the projection of the total spin on the z-axis is maximized. Write down the ground state wave function! 4 Perturbed harmonic oscillator A two dimensional harmonic oscillator has a degenerate eigenvalue E = 3¯ hω . What happens to this energy level due to the perturbation ˆ H 1 = C ˆ x ˆ y where C is a constant. 5 Time-dependent perturbation Assume that a system, described by a time-independent Hamiltonian ˆ H , is perturbed by ˆ H 1 ( t ) = ( ˆ H e t/τ for t < for t > , where ˆ H is small. At time t =-∞ the system is in an energy eigenstate | n i of ˆ H . Calculate, using first order time-dependent perturbation theory, the state of the....
View Full Document

## This note was uploaded on 12/12/2009 for the course FIZIK 201 taught by Professor Belmaşimsek during the Spring '09 term at Çukurova University.

### Page1 / 6

2 - TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK F ¨...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online