I Conceptual question about spin state rotations

[mindarson]

My question is conceptual but specific. I'm self-studying Townsend's text 'A Modern Approach to Quantum Mechanics.' In Sec. 2.2 pg 33 (in case you have the book handy), he introduces rotation operators, in the context of spin states for spin-1/2 particles.

He states that the rotation operator R(π/2j) (where j denotes the unit vector in the +y direction of an xyz coordinate system) rotates the [+z> ket (sorry about the clumsy notation) into the [+x> ket. He even illustrates this with a straightforward diagram of an xyz coordinate system.

In the text he states the following:

'The interaction of the magnetic moment of a spin-1/2 particle with the magnetic field causes the quantum spin state of the particle to rotate about the direction of the field as time progresses. In particular, if the magnetic field points in the y direction and the particle starts out in the state [+z>, the spin will rotate in the xz plane. At some later time the particle will be in the state [+x>.'

But here's what's driving me crazy: Isn't it true that these kets that are being rotated don't even live in the same space as the magnetic field? I.e. they are not spacetime vectors. In fact, Townsend himself, earlier in the text, made a point of emphasizing this point, that these kets don't represent states in ordinary 3D space but spin states in a different, 2D space altogether. So how/why is he now talking about these kets rotating around in ordinary space, relative to a magnetic field defined in ordinary space? A space where they don't even live?

 

[Jilang]

A vector can precess whilst the z component of it can remain constant.

 

[mindarson]

A vector can precess whilst the z component of it can remain constant.

My question is why it makes sense to talk about two-dimensional spin-state vectors rotating in ordinary space as if they were ordinary position vectors. Forgive me, but I can't see how your reply helps with that. Can you expound any further?

 

[PeroK]

But here's what's driving me crazy: Isn't it true that these kets that are being rotated don't even live in the same space as the magnetic field? I.e. they are not spacetime vectors. In fact, Townsend himself, earlier in the text, made a point of emphasizing this point, that these kets don't represent states in ordinary 3D space but spin states in a different, 2D space altogether. So how/why is he now talking about these kets rotating around in ordinary space, relative to a magnetic field defined in ordinary space? A space where they don't even live?

You have perhaps two separate issues here.

1) A physical rotation of a system may change the state ket. There ought to be a correspondence therefore between a physical rotation of a system and the "rotation" of the state ket.

This relation is part of the foundation of the theory of angular momentum. I don't have Townsend's book, so I don't know how and where he covers this.

For a spin-1/2 system, a physical rotation of $\phi$ about the z-axis results in a change to the state ket associated with the following state-space "rotation" operator:

$exp(\frac{-iS_z\phi}{\hbar})$

2) In the spin precession of a spin-1/2 system (with a magnetic field in the z-direction), the Hamiltonian is given by $\omega S_z$ and hence the time-evolution operator is:

$exp(\frac{-iS_z\omega t}{\hbar})$

Which is essentially the rotation operator (in ket space). Time evolution of this system, therefore, results in a physical 3D rotation of the system.

In summary, the physical rotation of the system and the "rotation" of the state ket are related, but as you know one is a physical 3D rotation and the other is the action of an operator in whatever ket space applies to the system.

 

[mikeyork]

A physical rotation is a unitary transformation in Hilbert space. In general it will take an eigenstate of $S_z$ into a superposition of $S_z$ and, in the case of a $\pi/2$ rotation about the $y$-axis, an eigenstate in $S_x$.

 

[PeterDonis]

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Edit: An off topic subthread has been deleted. Thread reopened.