What is the range of angles for the quantum spin correlation formula?

In summary, the conversation discusses the relationship between measuring spin in different directions and the probability of obtaining the same or opposite values. It is determined by the angle between the two axes and can be calculated using a rotation matrix. The range of angles for this formula is not clearly defined, but it is suggested that it may be associated with a 720° rotation for fermions.
  • #1
gametheory
6
0
So in my quantum class we learned that if you measure spin in one direction and get h/2 and then in another direction that it will be (plus or minus)h/2 as well. I was wondering how you would know the probability of it being the positive value vs the negative value. It's a function of the angle between the two directions, right?
 
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  • #2
I'm just a beginner myself, so wait for other replies. I've found that if spin is measured along a given axis of a spin½ particle, then the probability (p) that spin then measured along another axis will have the same sign (+ or -) is:

p = cos(α)/2 + 50%,

where α is the angle the new axis makes with the original axis.

Good luck!
 
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  • #3
That looks like it works. Do you know where this comes from? I figured it involves the pauli matrices in each of two orthogonal directions acting on the possible wavefunctions, but I wasn't sure how to arrange that into an equation.
 
  • #4
you want to look at stern-gerlach experiment i think. should be some good reading.
 
  • #5
Under a finite rotation, angular momentum states transform into each other with the aid of a unitary matrix. Naturally this matrix is called the rotation matrix, and written d(j)mm'(β) = (j m'|exp(iβ/ħ Jy)|j m) where β is the angle. For j = 1/2, d(j)mm'(β) = [tex]\left(\begin{array}{cc}cos(β/2) &sin(β/2)\\- sin(β/2)&cos(β/2)\end{array}\right)[/tex] Thus, if you have a state with Jz = +1/2 and describe it in terms of the spin states quantized along an axis inclined at an angle β to the z axis, the probability amplitude of finding Jβ = +1/2 is cos(β/2) and the probability amplitude of finding Jβ = -1/2 is sin(β/2).
 
  • #6
gametheory said:
That looks like it works. Do you know where this comes from?

I'm afraid I made mine up from a table of some spin correlations.
 
  • #7
I think that:

Cos^2(theta/2) = Cos(theta)/2 + .5

Where theta = {0, 45, 60, 90 degrees} that looks pretty good. :smile:
 
  • #8
Interesting! My daughter's old precalculus book lists the "Double Angle Formulas":

cos = 2cos2θ -1 = 1 - 2sin2θ = cos2θ - sin2θ, also

sin = 2sinθcosθ

This raised a follow-up question, if I may. (I'll start a new thread if needed.)
Over what range of angles is the quantum spin correlation formula considered complete?

The table I had listed five spin correlations from 0° to 180°. Classical trigonometry addresses 360° in its unit circle. But it seems that quantum spin is associated with a 720° rotation, at least for fermions.
 
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What is the Second Measurement of Spin?

The Second Measurement of Spin is a physical concept in quantum mechanics that describes the intrinsic angular momentum of a particle, such as an electron or proton.

Why is the Second Measurement of Spin important?

The Second Measurement of Spin is important because it is a fundamental property of particles that helps us understand their behavior and interactions in the quantum world. It also has important applications in technologies such as magnetic resonance imaging (MRI) and quantum computing.

How is the Second Measurement of Spin measured?

The Second Measurement of Spin is measured using a specialized tool called a spin detector, which can detect the direction of the spin of a particle. This measurement is then used to determine the spin state of the particle.

What is the difference between the First and Second Measurement of Spin?

The First and Second Measurement of Spin are two different properties of particles. The First Measurement of Spin, also known as spin-1/2, describes the spin of particles in terms of up and down states. The Second Measurement of Spin, also known as spin-1, describes the spin of particles in terms of three possible states: up, down, and sideways.

What are the implications of the Second Measurement of Spin on our understanding of the universe?

The Second Measurement of Spin, along with other concepts in quantum mechanics, challenges our traditional understanding of the universe and shows that particles can have properties that are not easily explained by classical physics. It also has implications for our understanding of quantum entanglement and the nature of reality at a fundamental level.

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