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  • Angle bisection

    From Stefan Ram@21:1/5 to All on Sat Sep 24 13:11:14 2022
    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?

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  • From Julio Di Egidio@21:1/5 to Stefan Ram on Sun Sep 25 11:30:43 2022
    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
    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

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