130_notes.dvi

5. Find the (normalized) eigenvectors and eigenvalues of theSx(matrix) operator fors= 1 in
   the usual (Sz) basis.

6.*A spin^12 particle is in a magnetic field in thexdirection giving a HamiltonianH=μBBσx.

Find the time development (matrix) operatore−iHt/ ̄hin the usual basis. Ifχ(t= 0) =

(

1

0

)

,

findχ(t).

7. A spin^12 system is in the following state in the usualSzbasis:χ=√^15

(√

3

1 +i

)

. What is the

probability that a measurement of thexcomponent of spin yields +^12?

8. A spin^12 system is in the stateχ=√^15

(

i

2

)

(in the usualSzeigenstate basis). What is the

probability that a measurement ofSxyields− 2 h ̄? What is the probability that a measurement

ofSyyields− 2 ̄h?

9. A spin^12 object is in an eigenstate ofSy with eigenvalue ̄h 2 at t=0. The particle is in a
   magnetic fieldB= (0, 0 ,B) which makes the Hamiltonian for the systemH=μBBσz. Find
   the probability to measureSy= ̄h 2 as a function of time.


10. Two degenerate eigenfunctions of the Hamiltonian are properlynormalized and have the fol-
    lowing properties.

Hψ 1 =E 0 ψ 1 Hψ 2 =E 0 ψ 2

Pψ 1 =−ψ 2 Pψ 2 =−ψ 1

What are the properly normalized states that are eigenfunctions of H and P? What are their

energies?

11. What are the eigenvectors and eigenvalues for the spin^12 operatorSx+Sz?


12. A spin^12 object is in an eigenstate ofSy with eigenvalue ̄h 2 at t=0. The particle is in a
    magnetic fieldB= (0, 0 ,B) which makes the Hamiltonian for the systemH=μBBσz. Find
    the probability to measureSy= ̄h 2 as a function of time.


13. A spin 1 system is in the following state, (in the usualLzeigenstate basis):

χ=

1

√

5





√i

2

1 +i



.

What is the probability that a measurement ofLxyields 0? What is the probability that a

measurement ofLyyields− ̄h?

14. A spin^12 object is in an eigenstate ofSz with eigenvalue ̄h 2 at t=0. The particle is in a
    magnetic fieldB= (0,B,0) which makes the Hamiltonian for the systemH=μBBσy. Find
    the probability to measureSz= ̄h 2 as a function of time.


15. A spin 1 particle is placed in an external field in theudirection such that the Hamiltonian is
    given by

H=α

(√

3

2

Sx+

1

2

Sy

)

Find the energy eigenstates and eigenvalues.