Probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?

In summary, the probability of measuring ##+\frac \hbar 2## along ##\hat n## is given by:P(+\frac \hbar 2 \, \, \hat n) = 1/2 (cos (\theta/2) e^{i\theta /2} |+ \rangle_n + sin (\theta/2)e^{i\theta /2} |- \rangle_n)
  • #1
happyparticle
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Homework Statement
Probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?
Relevant Equations
##|+\rangle_n = cos(\theta/2)e^{-i\phi/2}|+\rangle +sin(\theta/2)e^{i\phi/2}|-\rangle##
##|-\rangle_n = sin(\theta/2)e^{-i\phi/2}|+\rangle -cos(\theta/2)e^{i\phi/2}|-\rangle##
Hi,

Given a spin in the state ##|z + \rangle##, i.e., pointing up along the z-axis what are the probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?

My problem is that I'm not sure to understand the statement. It seems like I have to find the probabilities of measuring an eigenvalue along ##\hat{n}##. What does that mean exactly? Is it the probability to measure ##\pm \hbar/2## in the ##n## basis?

I tried to find ## |+\rangle ## in the ##n## basis, which I think this is ##|z+\rangle## in the ##n## basis. I thought maybe it could help me.
I got ## |+\rangle = 1/2 (cos (\theta/2) e^{i\theta /2} |+ \rangle_n + sin (\theta/2)e^{i\theta /2} |- \rangle_n)##

I'm really confuse with this statement. I'm not sure to understand the difference between ##|z + \rangle## and ##|+\rangle##
 
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  • #2
happyparticle said:
My problem is that I'm not sure to understand the statement. It seems like I have to find the probabilities of measuring an eigenvalue along ##\hat{n}##. What does that mean exactly?
The question is this. Imagine you have a system that produces electrons in the ##\ket{z+}## state. You can confirm this by setting up a measurement aparatus to measure the spin component about the z-axis and finding that every electron produced by your system behaves in the same way, corresponding to a measurement of ##+\frac \hbar 2##. Note that the z direction is simply some direction you have chosen in your experimental set-up.

Now, you leave your electron production system in place and re-orient your measurement aparatus along an axis ##\hat n##, represented by azimuthal and polar angles ##\theta, \phi##.

Each electron will behave in one of two ways, corresponding to the measurements of ##\pm \frac \hbar 2##. The question is: what is the probability that the electron is measured to have ##+\frac \hbar 2##; and what is the probability that the electron is measured to have ##-\frac \hbar 2##?
 
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Likes vanhees71 and topsquark

1. What is the significance of measuring ##\pm \hbar/2## along ##\hat{n}##?

The measurement of ##\pm \hbar/2## along ##\hat{n}## is significant because it represents the possible outcomes of a measurement of the spin of a particle in a specific direction. This measurement is related to the uncertainty principle in quantum mechanics and plays a crucial role in understanding the behavior of particles at the subatomic level.

2. How is the probability of measuring ##\pm \hbar/2## along ##\hat{n}## calculated?

The probability of measuring ##\pm \hbar/2## along ##\hat{n}## is calculated using the Born rule, which states that the probability of obtaining a particular measurement outcome is equal to the square of the amplitude of the corresponding quantum state component. In this case, the amplitude is related to the spin state of the particle in the direction of ##\hat{n}##.

3. Can the probability of measuring ##\pm \hbar/2## along ##\hat{n}## be greater than 1?

No, the probability of measuring ##\pm \hbar/2## along ##\hat{n}## cannot be greater than 1. According to the Born rule, the maximum probability of obtaining a particular measurement outcome is 1, which corresponds to a certainty of that outcome occurring. Any probability greater than 1 would violate the principles of quantum mechanics.

4. How does the choice of ##\hat{n}## affect the probabilities of measuring ##\pm \hbar/2##?

The choice of ##\hat{n}## affects the probabilities of measuring ##\pm \hbar/2## because it determines the direction in which the spin of the particle is measured. Different directions will have different probabilities, and the sum of all probabilities for all possible directions must equal 1.

5. What other factors can influence the probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?

The probabilities of measuring ##\pm \hbar/2## along ##\hat{n}## can also be influenced by external factors such as the presence of magnetic fields or interactions with other particles. These factors can affect the spin state of the particle and therefore alter the probabilities of obtaining a particular measurement outcome.

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