- #1
jeebs
- 325
- 4
Hi,
I have this problem on a past exam paper I am having some trouble with:
"in the conventional basis of the eigenstates of the Sz operator, the spin state of a spin-1/2 particle is described by the vector:
[tex] u = \left( \stackrel{cos a}{e^i^b sina} \right)[/tex] where a and B are constants.
find the probability that a measurement of the y-component of the spin of the particle will yield the result [tex] 0.5\hbar [/tex] ."
For the life of me I cannot work out how to write out matrices legibly on this thing, so I will summarize what is bothering me. I am given pauli matrices [tex] \sigma_x_,_y_,_z [/tex]that I cannot write out properly, and the spin operator is given by [tex] S_i = i\hbar\sigma_i [/tex].
In the question I am given the vector u, which is apparently expressed in the basis of Sz eigenstates.
Am I justified in putting this vector u into an eigenvalue equation [tex] S_{y} u = a_{y} u [/tex] ,
where ay is my eigenvalue, when the vector I would be operating on is made from a basis of eigenstates of another operator (Sz)?
I tried this and got two equations for ay, neither of which gives [tex] a_y = 0.5\hbar [/tex].
does this mean I can conclude that there is zero probability of finding the y-component of the spin being equal to [tex] 0.5\hbar [/tex] ?
or do I somehow have to wangle it so that I get another vector (not u) that is in the Sy eigenstate basis?
thanks.
PS. sorry, this crazy thing will not let me change something 5 lines up where I should have said
"Am I justified in putting this vector u into an eigenvalue equation S_y u = 0.5\hbar u"
I have this problem on a past exam paper I am having some trouble with:
"in the conventional basis of the eigenstates of the Sz operator, the spin state of a spin-1/2 particle is described by the vector:
[tex] u = \left( \stackrel{cos a}{e^i^b sina} \right)[/tex] where a and B are constants.
find the probability that a measurement of the y-component of the spin of the particle will yield the result [tex] 0.5\hbar [/tex] ."
For the life of me I cannot work out how to write out matrices legibly on this thing, so I will summarize what is bothering me. I am given pauli matrices [tex] \sigma_x_,_y_,_z [/tex]that I cannot write out properly, and the spin operator is given by [tex] S_i = i\hbar\sigma_i [/tex].
In the question I am given the vector u, which is apparently expressed in the basis of Sz eigenstates.
Am I justified in putting this vector u into an eigenvalue equation [tex] S_{y} u = a_{y} u [/tex] ,
where ay is my eigenvalue, when the vector I would be operating on is made from a basis of eigenstates of another operator (Sz)?
I tried this and got two equations for ay, neither of which gives [tex] a_y = 0.5\hbar [/tex].
does this mean I can conclude that there is zero probability of finding the y-component of the spin being equal to [tex] 0.5\hbar [/tex] ?
or do I somehow have to wangle it so that I get another vector (not u) that is in the Sy eigenstate basis?
thanks.
PS. sorry, this crazy thing will not let me change something 5 lines up where I should have said
"Am I justified in putting this vector u into an eigenvalue equation S_y u = 0.5\hbar u"
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