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A system is described by the following Hamiltonian:

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Quantum Mechanics September, 7 2017, I part

PROBLEM A

A system is described by the following Hamiltonian:

H = ¯ hω 2 −3i 3i −2

!

(1) Let us consider the following observables:

O

1

= 0 1 1 0

!

(2) and

O

2

= 0 −i i 0

!

(3) At the time t = 0 a measure of O

1

gives as a results its minimal eigenvalue. Find the probability that a measure of O

1

at the generic time t gives the same result in the three cases:

1. In the time interval [0, t] no measure has been performed on the system.

2. At the time t

1

∈ [0, t] we measure O

2

and as a result we get the maximal eigenvalue of O

2

.

3. At the time t

1

∈ [0, t] we measure O

2

.

PROBLEM B

Two particles with spin s

1

= 1 and s

2

= 2 are at rest in a configuration with total spin 3 and total spin projection S

z

= −¯ h. Which results can we obtain by measuring s

z2

? What are the probabilities associated with such measures?

PROBLEM C

An 1–d harmonic oscillator with frequency ω and mass m is perturbed by the potential V = ˆ ˆ p

2

Determine the energy shift for all eigenvalues to first order in .

1

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