Stern Gerlach expectation value

In summary, an electron with initial spin state |z ↑⟩ and Hamiltonian \hat{H} = αB0(\hat{S_x} +\hat{S_z}) evolves in a magnetic field and has an expectation value of -1/(1+√2) for the spin operator S_x at time t = 1/2. The energy eigenvalues and eigenvectors are E_+ = √2 and E_- = -√2 with corresponding eigenvectors |E_+⟩ and |E_-⟩, and the period of the electron's evolution is given by T = √2π/(αB0). To
  • #1
bmxicle
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Homework Statement


An electron starts in a spin state [tex]|\psi(t=0)\rangle = |z \uparrow \rangle [/tex] and evolves in a magnetic field [tex]B_0(\hat{x} + \hat{z})[/tex]. The Hamiltonian of the system is [tex]\hat{H} = \alpha \vec{B}\cdot\vec{S}[/tex]. Evaluate [tex] \langle \psi (t_{1/2}) | S_x | \psi(t_{1/2}) \rangle [/tex]

Homework Equations


[tex] S_x = \frac{\hbar}{2}\left( \begin{array}{cc}0 & 1\\ 1 & 0 \end{array} \right) [/tex]
[tex] S_z = \frac{\hbar}{2}\left( \begin{array}{cc}1 & 0\\ 0 & -1 \end{array} \right) [/tex]

The Attempt at a Solution


From another part of the question I found the period as [tex]T = \dfrac{\sqrt{2}\pi}{\alpha B_o}[/tex] since [tex] \hat{H} = \alpha B_o(\hat{S_x} +\hat{S_z}) [/tex]

The energy eigenvalues/eigenvectors are are [tex] E_+ = \sqrt{2} \rightarrow |E_+\rangle = \frac{1}{A} \left( \begin{array}{cc} 1 + \sqrt{2} \\ 1 \end{array} \right) [/tex]

[tex] E_- = -\sqrt{2} \rightarrow |E_-\rangle = \frac{1}{B} \left( \begin{array}{cc} 1 - \sqrt{2} \\ 1 \end{array} \right) [/tex]
[tex] A = (4+2\sqrt{2})^{1/2} \ and \ B = (4-2\sqrt{2})^{1/2} [/tex]

Which when plugging in the value for t_1/2 gives: [tex] |\psi(t_{1/2})\rangle =Ae^{-i\frac{\pi}{2}}|E_+\rangle + B e^{i\frac{\pi}{2}} |E_-\rangle = |\psi(t_{1/2})\rangle =-iA|E_+\rangle + iB|E_-\rangle [/tex]

I'm confused about changing basis to compute the expectation. Since the electron is initially in [tex] | z \uparrow \rangle [/tex] does this mean I want to change the expression I have for energy in terms in the eigenstates of S_x?
ie. Are the the S matrices all in the z-basis since that is what [tex] |\psi(t=0)\rangle[\tex] is given in?
 
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  • #2
So then I would have S_x = \frac{\hbar}{2}\left( \begin{array}{cc}0 & 1\\ 1 & 0 \end{array} \right) = \frac{1}{A}\left( \begin{array}{cc} A^2 & AB \\ AB & B^2 \end{array} \right) And the expectation value would be \langle \psi (t_{1/2}) | S_x | \psi(t_{1/2}) \rangle = \frac{iAB}{A^2B^2}(-iA^2 + iB^2) = \frac{AB}{A^2B^2}(B^2-A^2) = \frac{AB}{A^2B^2}(-2\sqrt{2}) = \frac{-2\sqrt{2}}{4+2\sqrt{2}}(-2\sqrt{2}) = -\frac{4}{4+2\sqrt{2}}
 

What is the Stern Gerlach expectation value?

The Stern Gerlach expectation value is a concept in quantum mechanics that describes the expected outcome of a measurement on a particle's spin in a magnetic field. It is the average value that would be obtained if the measurement was repeated many times on identical particles.

How is the Stern Gerlach expectation value calculated?

The Stern Gerlach expectation value is calculated using the probability distribution of the particle's spin states. It is the sum of the spin states, each multiplied by their respective probabilities.

What is the significance of the Stern Gerlach expectation value?

The Stern Gerlach expectation value is significant because it provides a way to make predictions about the behavior of particles in magnetic fields. It is also one of the first experiments to show the quantization of spin in quantum mechanics.

What factors can affect the Stern Gerlach expectation value?

The Stern Gerlach expectation value can be affected by the strength and direction of the magnetic field, the properties of the particles being measured, and the position and orientation of the measuring apparatus.

What are some real-world applications of the Stern Gerlach expectation value?

The Stern Gerlach expectation value has been used in many experiments and technologies, such as magnetic resonance imaging (MRI) and spintronics. It also plays a crucial role in understanding the behavior of particles in particle accelerators and quantum computers.

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