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A two level system is described by the Hamiltonian H = ¯ hω 4 3i

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Quantum Mechanics July, 6 2017

PROBLEM A

A two level system is described by the Hamiltonian H = ¯ hω 4 3i

−3i −4

!

. (1)

Initially is in the state |1i = 1 0

!

.

1. Determine eigenvalues and eigenvectors of H.

2. What is the probability to found the system at the time t in the state |2i = 0 1

!

? 3. Determine at what time t the probability to find the system in state |2i is 1/2.

PROBLEM B

Consider a particle with mass m and charge q in a 3d harmonic potential V (x, y, z) = 1

2 k(x

2

+ y

2

+ z

2

) ≡ 1 2 k|~ r|

2

Switch on a uniform magnetic field B along z. Neglecting the spin degrees of freedom the Hamiltonian is given by,

H = 1

2m (~ p)

2

+ 1

2 k|~ r|

2

− qB 2mc L

z

.

1. Re-write the Hamiltonian using creation and annihilation operators for the harmonic oscillator.

2. Using perturbation theory for small B find the smallest corrections for the first two energy levels.

1

Riferimenti

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