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Sep 24, 2022, 4:11:18 PM9/24/22

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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?

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?

Sep 25, 2022, 6:30:47 AM9/25/22

to

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
> 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>.

|-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>.

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
> 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.

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

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