- #1

phoenix3

- 7

- 0

- Homework Statement
- How to interpret superposition

- Relevant Equations
- ##cos^2(\frac{\theta}{2})##

Here is my workings out:

$$$$

If a particle's spin of magnitude ##\frac {\hbar}{2}## is prepared along direction ##\vec r_1## and subsequently its spin is measured along direction ##\vec r_2 ## at an angle ##\vec \theta ## to ##\vec r_1##, the probability of its being found "spin up" along is ##\vec r_2## is ##P(up)=cos^2(\frac{\theta}{2})## again with full magnitude ##\frac {\hbar}{2}##.

$$ $$

Classically, the spin might easily have been incorrectly predicted to be of magnitude ##\frac {\hbar}{2} . cos(\theta)## along ##\vec r_2 ##

$$ $$

in units of ##\frac {\hbar}{2}##, ##\frac {\hbar}{2} = 1##.

$$ $$

Starting with the seemingly 'incorrect' assumption that the component along ##\vec r_2 ## is ##\frac {\hbar}{2} . cos(\theta)##

$$ $$

$$$$

##\frac {\hbar}{2} . cos(\theta) = 1.cos(\theta) ## in units of ##\frac {\hbar}{2}##

\begin{align}

1.cos(\theta) & = cos(\frac{\theta}{2}+\frac{\theta}{2}) \nonumber \\

& = cos^2(\frac{\theta}{2})-sin^2(\frac{\theta}{2}) \nonumber \\

& = (+1).cos^2(\frac{\theta}{2}) + (-1).sin^2(\frac{\theta}{2}) \nonumber \\

\end{align}

Multiplying accross by a time interval, ##\delta t##,

\begin{align}

1.cos(\theta).\delta t = (+1).cos^2(\frac{\theta}{2}).\delta t + (-1).sin^2(\frac{\theta}{2}).\delta t \nonumber \\

\end{align}

it seems that this can be interpreted to mean either:

$$$$

(A) a combination of two vectors of magnitudes ## cos^2(\frac{\theta}{2}) ## and ##-sin^2(\frac{\theta}{2})## respectively, acting for duration ##\delta t## or

$$$$

(B) a combination of two vectors of magnitudes 1 and -1 respectively, the first acting for a duration ##cos^2(\frac{\theta}{2}).\delta t## and the second acting for the remaining duration ##sin^2(\frac{\theta}{2}).\delta t##

$$$$

with ##( cos^2(\frac{\theta}{2}).\delta t + sin^2(\frac{\theta}{2}).\delta t = \delta t )##

$$$$

This latter interpretation, (B), would seem to justify quite nicely why the spin is always measured to have the same absolute unit magnitude and also why the probability of getting "spin up" along ##\vec r_2 ## is equal to ##cos^2(\frac{\theta}{2})## since according to this interpretation it is in fact "spin up" along ##\vec r_2## , ## cos^2(\frac{\theta}{2})## of the time.

$$$$

Obviously once measured along ##\vec r_2 ## it is then in an eigenstate along ##\vec r_2 ## and the probability of a measurement along ##\vec r_1 ## being "spin up" along ##\vec r_1 ## is ##cos^2(\frac{\theta}{2})## for the same reason.

$$$$

Is there an error in the above and if so where?

$$$$

If a particle's spin of magnitude ##\frac {\hbar}{2}## is prepared along direction ##\vec r_1## and subsequently its spin is measured along direction ##\vec r_2 ## at an angle ##\vec \theta ## to ##\vec r_1##, the probability of its being found "spin up" along is ##\vec r_2## is ##P(up)=cos^2(\frac{\theta}{2})## again with full magnitude ##\frac {\hbar}{2}##.

$$ $$

Classically, the spin might easily have been incorrectly predicted to be of magnitude ##\frac {\hbar}{2} . cos(\theta)## along ##\vec r_2 ##

$$ $$

in units of ##\frac {\hbar}{2}##, ##\frac {\hbar}{2} = 1##.

$$ $$

Starting with the seemingly 'incorrect' assumption that the component along ##\vec r_2 ## is ##\frac {\hbar}{2} . cos(\theta)##

$$ $$

$$$$

##\frac {\hbar}{2} . cos(\theta) = 1.cos(\theta) ## in units of ##\frac {\hbar}{2}##

\begin{align}

1.cos(\theta) & = cos(\frac{\theta}{2}+\frac{\theta}{2}) \nonumber \\

& = cos^2(\frac{\theta}{2})-sin^2(\frac{\theta}{2}) \nonumber \\

& = (+1).cos^2(\frac{\theta}{2}) + (-1).sin^2(\frac{\theta}{2}) \nonumber \\

\end{align}

Multiplying accross by a time interval, ##\delta t##,

\begin{align}

1.cos(\theta).\delta t = (+1).cos^2(\frac{\theta}{2}).\delta t + (-1).sin^2(\frac{\theta}{2}).\delta t \nonumber \\

\end{align}

it seems that this can be interpreted to mean either:

$$$$

(A) a combination of two vectors of magnitudes ## cos^2(\frac{\theta}{2}) ## and ##-sin^2(\frac{\theta}{2})## respectively, acting for duration ##\delta t## or

$$$$

(B) a combination of two vectors of magnitudes 1 and -1 respectively, the first acting for a duration ##cos^2(\frac{\theta}{2}).\delta t## and the second acting for the remaining duration ##sin^2(\frac{\theta}{2}).\delta t##

$$$$

with ##( cos^2(\frac{\theta}{2}).\delta t + sin^2(\frac{\theta}{2}).\delta t = \delta t )##

$$$$

This latter interpretation, (B), would seem to justify quite nicely why the spin is always measured to have the same absolute unit magnitude and also why the probability of getting "spin up" along ##\vec r_2 ## is equal to ##cos^2(\frac{\theta}{2})## since according to this interpretation it is in fact "spin up" along ##\vec r_2## , ## cos^2(\frac{\theta}{2})## of the time.

$$$$

Obviously once measured along ##\vec r_2 ## it is then in an eigenstate along ##\vec r_2 ## and the probability of a measurement along ##\vec r_1 ## being "spin up" along ##\vec r_1 ## is ##cos^2(\frac{\theta}{2})## for the same reason.

$$$$

Is there an error in the above and if so where?

Last edited: