Spin 1/2 planced in a magnetic field at x-direction

In summary, a spin-1/2 placed in a magnetic field at x-direction has a Hamiltonian of \hat{H} = -BSx, where Sx = h/2 * pauli x-matrix. The energy eigenvalues are \lambda = \pm \frac{B hbar}{2} and the corresponding eigenvectors are [1 1] and [1 -1]. The lowest energy state is achieved when Sx has an eigenvalue of hbar/2, and the probability of measuring hbar/2 in this state is 1.
  • #1
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spin 1/2 placed in a magnetic field at x-direction

Hello guys
I have been a regular reader of this forum. This is my first thread.

Homework Statement



A spin-1/2 is place in a magnetic field which points in the x-direction. The Hamiltonian is [tex]\hat{H}[/tex] = - B Sx

where Sx = [STRIKE]h[/STRIKE]/2 * pauli x-matrix. We have absorbed all couplings into an effective magnetic field B>0 so that B has dimension inverse time.

1. Find the energy eigenvalues and corresponding eigenstates of H.

the spin is known to be in its lowest energy state. Then a measurement (M1) of the spin-component in the x-direction is carried out.

2. Calculate the probability of finding the value h-bar /2 in measurement M1.

The Attempt at a Solution



Many attempts but, don't have a clue where to start from.
 
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  • #2
To start with: Can you write [tex]S_x$[/tex] as a matrix? What matrix would it be? Can you find eigenvectors and eigenvalues of this matrix?
 
  • #3
yes. the S_x matrix is (h-bar)/2 * pauli x- matrix. I have written it in the question above.
 
  • #4
Can you find eigenvectors and eigenvalues of this matrix?
 
  • #5
yes. the eigenvalues are +- (h-bar/ 2). and the eigenvectors are [1 1] and [1 -1]

So the eigenvalues of Hamiltonian become :[tex]\lambda[/tex] = [tex]\pm[/tex] [tex]\frac{B hbar}{2}[/tex]
the eigenvectors are the same as written above.

what about the propability now?
 
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  • #6
So, which will be "the lowest energy eigenstate"?
 
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  • #7
Since the hamiltonian has two eigenvalues, so the one with lowest energy will be E = (- Hbar*B)/2 ? or is it the plus one. ?
 
  • #8
Alright, the minus one is lower. Now, if you want to calculate probabilities, you must normalize the eigenvector. What will be the normalized (norm one) eigenvector?
 
  • #9
The normalized eigenvectors: [tex]\frac{1}{\sqrt{2}}[/tex] [1 1]. And the probability for getting hbar/2 is 0?
 
  • #10
Now, be careful. You have H=-BSx. Eigenvalues of Sx are hbar/2 and -hbar/2. When Sx has eigenvalue hbar/2, H has eigenvalue -Bhbar/2 - which is the lowest. In general, the probability is given by the expression of the form

[tex]|<u|v>|^2[/tex]

But here |u> and |v> are the same: on your (normalized) eigenvector u=v= [tex]
\frac{1}{\sqrt{2}}
[/tex][1,1] H has the lowest eigenvalue and Sx has eigenvalue hbar/2. So?
 
  • #11
i got it now. then the probability is 1. thanks
 
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Related to Spin 1/2 planced in a magnetic field at x-direction

1. What is spin 1/2?

Spin 1/2 refers to the intrinsic angular momentum of a particle, which is a fundamental property of particles in quantum mechanics. It is a quantum number that can take on values of either +1/2 or -1/2.

2. What is a magnetic field?

A magnetic field is a region in space where a magnetic force can be detected. It is created by moving electric charges and is characterized by its direction and strength.

3. What does it mean for a particle to be "placed" in a magnetic field at the x-direction?

When a particle is placed in a magnetic field at the x-direction, it means that the magnetic field is acting on the particle in the x-direction. This can be achieved through the use of a magnet or by altering the path of the particle to pass through the x-direction of the magnetic field.

4. How does a magnetic field affect a spin 1/2 particle?

A magnetic field will cause a spin 1/2 particle to precess, or rotate, around the direction of the magnetic field. This is due to the interaction between the magnetic field and the intrinsic magnetic moment of the particle, which is related to its spin.

5. What are the applications of studying spin 1/2 particles in a magnetic field?

Studying spin 1/2 particles in a magnetic field has many applications in fields such as quantum computing, materials science, and medical imaging. It can also help us better understand the fundamental properties of particles and the behavior of matter at a microscopic level.

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