Angle bisection

[Stefan Ram]

I'm just watching Lecture 2 of "The Theoretical Minimum:
Quantum Mechanics" by Leonard Susskind, about 1 hour and
10 minutes in.

A: IIRC, the state |spin up> is orthogonal to the state |spin down>,
because if one prepares a source for |spin up>, one never measures
|spin down>.

B: If one prepares a source of |spin up>, one always measures
|spin up>. But if one rotates the measurement device by 180 degrees
so that it is upside down, one always measures |spin down>.

So, the state |spin down> in the Hilbert space is orthogonal
to the state |spin up> (A). Usually, in the normal two- or three-
dimensional spaces I imagine that "orthogonal" means "90 degree".
But to get from |spin up> to |spin down> the measurement device has
to be rotated by "180 degrees" (B). It's as if the angle in the
Hilbert space of states (90 degrees) is /half the angle/ (180 degrees)
in locational space.

Then, Susskind talks about the states |spin left> and |spin right>
one measures by rotating the measurement device by 90 degrees.
He explains that |spin right> is (1/sqrt(2))|spin up>+
(1/sqrt(2))|spin down>. But I know that (1/sqrt(2))(1,1) are the
coordinates of a unit vector that encloses an angle of 45 degrees
with the x axis. So a rotation of 90 degrees in locational space
now corresponds to a rotation of 45 degrees in state space, again
a half of the angle of 90 degrees.

Finally, I remember vaguely that there is a situation where
the state is restored only after a rotation by 720 degrees
in the locational space, which by a bisection would correspond
to a rotation by 360 degrees (i.e., identity) in space state.
(It is difficult to imagine that after a rotation of 360 degrees
in locational space not everything is the same again!)

So, have I made a mistake in my description or has this been
observed and discussed before that sometimes a rotation in
locational space corresponds to half that rotation in state space?

[Julio Di Egidio]

On Saturday, 24 September 2022 at 22:11:18 UTC+2, Stefan Ram wrote:
> I'm just watching Lecture 2 of "The Theoretical Minimum:
> Quantum Mechanics" by Leonard Susskind, about 1 hour and
> 10 minutes in.
>
> A: IIRC, the state |spin up> is orthogonal to the state |spin down>,
> because if one prepares a source for |spin up>, one never measures
> |spin down>.

It's not dependent on preparation, spin |+1> is orthogonal to spin
|-1> because they are mutually exclusive in *measurement*: IOW,
when we measure the spin, in any spatial direction (!), we just get
either of two distinct *outcomes*, |+1> ("the electron did NOT give
off a photon") or |-1> ("the electron did give off a photon"), and
that is what makes them *orthogonal states*. (Notice that here
I am not saying |+1> and |-1>in any specific spatial sense, I am
using those as generic labels for the two and only two possible
distinct outcomes.)

> B: If one prepares a source of |spin up>, one always measures
> |spin up>. But if one rotates the measurement device by 180 degrees
> so that it is upside down, one always measures |spin down>.

No, for a spin prepared in the |up> direction, if we measure |down>,
we definitely get a |-1>, i.e. "NOT in that direction", which in this
case is in fact a definite |up>, i.e. the opposite of |down>. The point
is, when we prepare a spin, *whichever the direction*, the result of
measurement *in that direction* is definitely |+1>, and the result *in
the opposite direction* is definitely |-1> (i.e. however |+1> and |-1>
are concretely represented in the chosen basis).

<spin>

> But to get from |spin up> to |spin down> the measurement device has
> to be rotated by "180 degrees" (B). It's as if the angle in the
> Hilbert space of states (90 degrees) is /half the angle/ (180 degrees)
> in locational space.

It is true that we must rotate a spin by 720° to get back to the
same state, because a rotation by 360° gives back the initial
amplitudes negated, so the observable spin is the same still a
difference remains detectable e.g. in interference, where the
quantum phase matters. This is explained better and in more
detail here: <https://en.wikipedia.org/wiki/Spin-1/2#Complex_phase>

But I think the point is you may be conflating the ordinary space
in which we prepare and measure with state space, and the two
are quite distinct. E.g. the state space for "spin (in any direction)"
is 2-dimensional because there are two and only two possible
outcomes of any spin measurement; OTOH, the state space for
"position (along some axis)" is infinitely dimensional since we
measure infinitely many possible different and distinct outcomes.

That said, please take that as just a first approximation: but even
Susskind, as he himself reminds the audience, at that point is still
doing informal introduction and exploration....

HTH,

Julio