Half Spin in QM: Calculating Probability of Measured Eigenvalue at Time T=2T

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In summary, the problem involves a spin with an eigenstate of the operator S_x at time t=0, and a magnetic field (0,0,B) corresponding to the Hamiltonian H=w(B)\hbar*\sigma_z. At time T, the magnetic field direction changes to (0,B,0) and after another time T, a measurement of S_x is done. The question is about the probability that the measured value is the same as the starting value. The approach involves calculating the probability using the operator and the wave function at different times. The final result is cos^2(2wT), with a possible factor of 0.25 depending on the calculations.
  • #1
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Problem:
a half spin has an eigenstate of the opertaor S_x (which is defined by the multiplication of half h bar times pauli sigma x matrix) of eigen value + half h bar at time t=0.
the spin is at a magnetic field (0,0,B) which correspond to the hamiltonian [tex]H=w(B)\hbar*\sigma_z[/tex], at time T they change the direction of the magnetic field to the y direction: (0,B,0), after another time T a measurement of S_x was done, what is the probability that the value measured is the one we started with?

My answer:
now from 0<t<T we have that [tex]|\psi(t)>=e^{-iHt/\hbar}|\psi(0)>[/tex]
which equals: [tex]|\psi(t)>=\frac{1}{\sqrt 2}(cos(wt)-isin(wt),cos(wt)+isin(wt))[/tex]
now from T to 2T we have a magnetic field working in the y direction, does it mean we should act the above operator on |psi(T)> but with the appropiate change i.e should it be soemthing like this, at time t=T, [tex]|psi(T)>=\frac{1}{\sqrt 2}(cos(wT)-isin(wT),cos(wT)+isin(wT))[/tex], now in order to find |psi(t)> at [T,2T] should it be:
[tex]|\psi(t)>=e^{-iw(B)\hbar \sigma_y(t-T)/\hbar}|\psi(T)>[/tex] or something else?
from there in order to calculate the wanted probability i need to compute:
[tex]||<\psi(0)|\psi(2T)>|^2[/tex]
is my approach correct or does it have loopholes?

any input?
thanks in advance.
 
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  • #2
anyone?
if someone already asked such a question please do point me to his thread.
 
  • #3
okay with my approach i got that that the probability is cos^2(2wT), is correct or not, i don't know, do you?
 
  • #4
looks ok.
 
  • #5
Well I think I miss a factor of 0.25, there.
 
  • #6
my bad, I got through my calculuations and i don't think there's missing a factor.
 

1. What is half spin in quantum mechanics?

Half spin refers to the intrinsic angular momentum of a particle, which is a fundamental property in quantum mechanics. It is represented by the quantum number s = 1/2 and is used to describe the spin of particles such as electrons and protons.

2. What is meant by "calculating probability of measured eigenvalue"?

In quantum mechanics, eigenvalues are the possible outcomes of a measurement of a physical property, such as spin. The probability of obtaining a specific eigenvalue is calculated using the mathematical concept of eigenstates, which represent the different possible states of a particle.

3. How is the probability of a measured eigenvalue calculated in the context of half spin at time T=2T?

The probability of a measured eigenvalue can be calculated using the time-dependent Schrödinger equation, which describes the evolution of a quantum system over time. By solving this equation, the probability of obtaining a specific eigenvalue at a given time can be determined.

4. What is the significance of T=2T in this context?

In quantum mechanics, the time parameter T represents the time at which a measurement is taken. By setting T=2T, we are looking at the probability of obtaining a specific eigenvalue at a time that is twice the initial time, which allows us to study the time evolution of the system and make predictions about its behavior.

5. How does half spin in QM relate to real-world applications?

Half spin and quantum mechanics in general have numerous real-world applications, such as in the development of technologies like transistors and lasers. It also plays a crucial role in fields like nuclear physics, materials science, and chemistry, allowing us to understand and manipulate the behavior of particles at the atomic and subatomic level.

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