Sakurai page 27: What is this 'delta_1'?

[omoplata]

In page 26 of 'Modern Quantum Mechanics (Revised edition)' by J.J. Sakurai, equation (1.4.9), they find the $S_x +$ ket,
$$\mid S_x;+ \rangle = \frac{1}{\sqrt{2}} \mid + \rangle + \frac{1}{\sqrt{2}} e^{i \delta_1} \mid - \rangle$$ with $\delta_1$ real. What is this $\delta_1$?

Furthermore, equation (1.4.8) says,
$$| \langle + \mid S_x ; + \rangle | = | \langle - \mid S_x ; + \rangle | = \frac{1}{\sqrt{2}}$$
But if you take equation(1.4.9) and apply $\langle - \mid$ to it from the left, it becomes,
$$\langle - \mid S_x ; + \rangle = \frac{1}{\sqrt{2}} e^{i \delta_1}$$
This does not match with equation (1.4.8). There's this extra factor of $e^{i \delta_1}$

Could someone help me with this? Thanks a lot.

 

[omoplata]

Please let me know if the question is not clear. I was assuming that almost everyone has Sakurai with them. If it's not clear I'll try to provide more context.

 

[Bill_K]

He says the two components of the beam have equal intensities. However they can differ in phase, and δ₁ is the relative phase.

Regarding your second question, note that Eq (1.4.8) has an absolute value sign, and |(1/√2)e^iδ1| = (1/√2).

 

[omoplata]

Oh cool! I forgot about the absolute value! Thanks!

I would like to ask one more question. What is this 'phase'. I don't really understand it. As far as I know, waves have a phase. In the case given in the book (Stern-Gerlach experiment), is the spin state ket (I think 'spin state ket' is that's what this $\mid S_x ; \pm \rangle$ means?) of the beam of particles oscillating in some way? Is there any similar simple observable model that I can connect with this, to understand this in a simpler way?

 

[LeonhardEuler]

Oh cool! I forgot about the absolute value! Thanks!

I would like to ask one more question. What is this 'phase'. I don't really understand it. As far as I know, waves have a phase. In the case given in the book (Stern-Gerlach experiment), is the spin state ket (I think 'spin state ket' is that's what this $\mid S_x ; \pm \rangle$ means?) of the beam of particles oscillating in some way? Is there any similar simple observable model that I can connect with this, to understand this in a simpler way?

The absolute phase is not observable. Only the effects of relative phase differences can be observed. For example, if you have any equation in quantum mechanics where i appears, you can replace it everywhere with -i (including implicitly in definitions) and end up with an equally valid equation (predictions about observables will be the same).

 

[omoplata]

The absolute phase is not observable. Only the effects of relative phase differences can be observed. For example, if you have any equation in quantum mechanics where i appears, you can replace it everywhere with -i (including implicitly in definitions) and end up with an equally valid equation (predictions about observables will be the same).

OK, thanks.

The only time I've encountered 'phase' before was in waves. If there is a 'phase', does that mean there is a 'frequency' as well?

 

[LeonhardEuler]

OK, thanks.

The only time I've encountered 'phase' before was in waves. If there is a 'phase', does that mean there is a 'frequency' as well?

Yes, it does when you are talking about time dependence. When you look at stationary states of the Schrodinger equation, which are states where nothing observable varies with time, the phase still varies with time according to the prefactor
$$e^{\frac{iEt}{\hbar}}$$
Since the phase is not observable, it is still the case that every observable is independent of time.

Interestingly, even the absolute frequency is not really observable even though in the equation it appears as a constant times the energy. This 'frequency' has the strange property that it changes depending on how you consider a system.

In classical mechanics, if you have two oscillating systems, S₁ and S₂ isolated from each other with energies E₁ and E₂, and frequencies ω₁ and ω₂, then when you consider them together they have energy E₁+E₂, while they still have separate frequencies ω₁ and ω₂.

In quantum mechanics, if you have systems with energies E₁ and E₂, you still have the total energy when the two systems are considered as one big system to be E₁+E₂. The strange thing is when you look at the phase frequencies of the separate systems, they are
$$ω_{1}=\frac{E_1}{\hbar}$$
$$ω_{2}=\frac{E_2}{\hbar}$$
But if you consider them together, the phase frequency is
$$ω_{TOT}=\frac{E_1+E_2}{\hbar}$$
This sort of thing discouraged people in the early days of quantum mechanics from looking for a physical interpretation of the phase and phase frequency. People now mostly just consider them some mathematical quantities that come out of the equations.

 

[omoplata]

Very interesting. So if we can't observe the phase or the frequency, then how do we know that they exist? Is there no way the mathematical foundations of quantum mechanics can be built without them?