Sakurai, Chapter 1 Problems 23 & 24

[quantumkiko]

Problem 23:
If a certain set of orthonormal kets, $$|1> |2> |3>$$, are used as the base kets, the operators A and B are represented by

$$A = \left( \begin{array}{ccc} a & 0 & 0 \\ <br /> 0 & -a & 0 \\ <br /> 0 & 0 & -a \end{array} \right)<br /> <br /> B = \left( \begin{array}{ccc} b & 0 & 0 \\ <br /> 0 & 0 & -ib \\ <br /> 0 & ib & 0 \end{array} \right).<br />$$

A and B commute. Find a new set of orthonormal kets which are simultaneous eigenkets of both A and B. Specify the eigenvalues of A and B for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket?


Problem 24:
Prove that $$(1 / \sqrt{2})(1 + i\sigma_x)$$ acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle $$-\pi / 2$$. (The minus sign signifies that the rotation is clockwise.)

 

[tiny-tim]

Hi quantumkiko!

Show us what you've tried, and where you're stuck, and then we'll know how to help.

 

[quantumkiko]

Hi Tim!

In problem 23, I don't know how to represent the simultaneous eigenkets $$|a, b>$$. I just know how to solve the eigenvalues for each operator using the characteristic equation (some are degenerate). I also know that for two commuting observables, their simultaneous eigenkets form a complete set. Therefore, their simultaneous eigenkets are automatically orthogonal. That's all.

For problem 24, I think we have to show that the result of letting the operator $$(1 / \\sqrt{2})(1 + i\\sigma_x)$$ act on a spinor is equivalent to a rotation operator acting on the same spinor. For a spinor of unit length, I used the matrix representation $$\left( \begin{array}{c} \cos \theta & \sin\theta \end{array} \right)$$ (I think this is where I was wrong.) Since the angle of rotation is $$-\pi / 2$$, the rotation matrix will be given by,

$$\left( \begin{array}{cc} cos(-\pi / 2) & sin(-\pi / 2) \\ -sin(-\pi / 2) & cos(-\pi / 2) \end{array} \right) = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$$.

If I let this operator act on the spinor, the resulting s

 

[tiny-tim]

Hint: in problem 23, just look at the bottom right-hand 2x2 square of A …

it's a multiple of the unit matrix!

so its eigenkets are … ?

 

[quantumkiko]

Hint: in problem 23, just look at the bottom right-hand 2x2 square of A …

it's a multiple of the unit matrix!

so its eigenkets are … ?

It's eigenkets are $$\left( \begin{array}{c} 1 \\ 1 \end{array}\right) and \left( \begin{array}{c} -1 \\ -1 \end{array}\right)$$?

 

[tiny-tim]

It's eigenkets are $$\left( \begin{array}{c} 1 \\ 1 \end{array}\right) and \left( \begin{array}{c} -1 \\ -1 \end{array}\right)$$?

waah!

think … if C is the 2x2 unit matrix,

for what vectors or kets V is CV = V?

 

[quantumkiko]

Oh, for all kets V! So how does that fit into finding the simultaneous eigenstates of A and B?

 

[tiny-tim]

Oh, for all kets V! So how does that fit into finding the simultaneous eigenstates of A and B?

Well, there's one obvious simultaneous eigenstate …

and once you've found the other two eigenstates of B, they're bound to be eigenstates of A also.

(i'm logging out now for a few hours )

 

[quantumkiko]

I got it! The obvious one is $$\left( \begin{array}{c} 1 & 0 & 0 \end{array} \right)$$ while the others are $$\left( \begin{array}{c} 0 & 1/\sqrt{2} & i/\sqrt{2} \end{array} \right)$$ and $$\left( \begin{array}{c} 0 & -1/\sqrt{2} & -i/\sqrt{2} \end{array} \right)$$. Thank you very much!

Now how about Problem # 24?

 

[tiny-tim]

…while the others are $$\left( \begin{array}{c} 0 & 1/\sqrt{2} & i/\sqrt{2} \end{array} \right)$$ and $$\left( \begin{array}{c} 0 & -1/\sqrt{2} & -i/\sqrt{2} \end{array} \right)$$.

erm … they're the same!

Now how about Problem # 24?

Le'ssee …

Problem 24:
Prove that $$(1 / \sqrt{2})(1 + i\sigma_x)$$ acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the x-axis by angle $$-\pi / 2$$. (The minus sign signifies that the rotation is clockwise.)

Well … to prove it's a π/2 rotation …

the obvious thing to do is to square it!

 

[quantumkiko]

Oh yeah, I should really get different eigenkets, not just multiples of one of the other. So the other two should be $$\left( \begin{array}{c} 0 & 1/\sqrt{2} & i/\sqrt{2} \end{array} \right)$$ and $$\left( \begin{array}{c} 0 & i/\sqrt{2} & 1/\sqrt{2} \end{array} \right)$$
I was thinking that they won't be orthonormal, but I forgot that one of the $$i$$'s changes sign when doing the inner product.

I got Problem # 24 also. Thank you!