Sterngerlach experiments problems,

[belleamie]

HI there, I was assigned 7 homework problems but there were three I didnt know how to answer...
please help, any hints on how to start would be appreciated.


#3 the state of spin-1/2 particle that is spin up along the axis whose direction is specified by the unit vector n=sin (theta) cos (phi) i+sin (theta) sin (phi)j+cos (theta)k, with theata and phi shown in attachment given by
|+n> = cos (theta/2)|+z>+e^(i*theta) sin (theta/2)|-z>

a) Verify that the state |+n> reduces to the states |+x> and |+y> for angles theta and phi

b)Suppose that a measurement of S(sub z) is carried out on a particle in the state |+n> What is the probability that the measurement yields ((hbar)/2)? and ((-hbar)/2))

c) Determine the uncertainty (change of S(subz))of your measurements


#7 a) what is the amp to find a particle that is in the state |+n> from problem #3 with S(sub y)=hbar/2? what is the probability? check result by evaluating he probability for an appropriate chocice of hte angles phi and theta
b)What is the amp to find a particle that is in the state |+y> with S(sub n)=hbar/2? What is hte probabtility?


#8 Show that the state
|+n> = sin(theta/2)|+z>-e^(i(theta)) cos (theta/2)|-z>
satisfies <+n|-n>=0, where the state |+n> is given from #3 Verify that <-n|-n>=1

 

[Norman]

First of all, you need to read the thread at the top of the page (of the homework help section) about guidelines for posting homework help. With that said I have these questions for you: What have you done so far? Are we supposed to answer these questions straight away for you? I sincerely doubt anyone will. Either way, we cannot help you unless we understand why you don't understand the problems and where you are getting stuck. Please post what you have done so far. Also make sure that problem number eight is written correctly.
Cheers,
Ryan


edit: didn't realize what forum I was in- my apologies

 

[quicknote]

Dear student,

Thank you for reaching out for help with your homework problems. I understand that you are struggling with three problems and are looking for some hints to get started. I am happy to provide some guidance and help you understand the concepts involved in these problems.

Problem #3:

a) To verify that the state |+n> reduces to the states |+x> and |+y>, you can use the given expression for |+n> and substitute the values of theta and phi for each case. For example, for |+x>, theta = pi/2 and phi = 0, so you can plug these values into the expression and see if it simplifies to |+x>. Similarly, for |+y>, theta = pi/2 and phi = pi/2. If the expressions simplify to the respective states, then you have verified the reduction.

b) To determine the probability of measuring a spin of hbar/2 or -hbar/2, you can use the formula P = |<n|+z>|^2 for the corresponding state. For example, for a spin of hbar/2, you would use the state |+z>. To find the uncertainty in the measurement of S(subz), you can use the formula sqrt(<S(subz)^2>-<S(subz)>^2).

Problem #7:

a) To find the amplitude to measure a particle in the state |+n> with S(suby) = hbar/2, you can use the formula <n|+y> and substitute the values of theta and phi for the given state |+n>. To find the probability, you can use the formula P = |<n|+y>|^2. You can then check your result by evaluating the probability for an appropriate choice of theta and phi.

b) Similarly, to find the amplitude and probability for a particle in the state |+y> with S(subn) = hbar/2, you can use the formula <y|+n> and substitute the values of theta and phi for the given state |+y>.

Problem #8:

To show that <+n|-n>=0, you can use the given expressions for |+n> and |-n> and substitute the values of theta and phi. If the result is 0, then you have shown that <+n|-n>=0. To verify that <-n|-n>=1