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Spin-1 Eigenvectors

eigenvectors Quantum Fundamentals 2023 The operator \(\hat{S}_x\) for spin-1 may be written as: defined by: \[\hat{S}_x=\frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0&1&0\\ 1&0&1 \\ 0&1&0 \\ \end{pmatrix} \]

1. Find the eigenvalues and eigenvectors of this matrix. Write the eigenvectors as both matrices and kets.
2. Confirm that the eigenstates you found give probabilities that match your expectation from the Spins simulation for spin-1 particles.

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Spin-1/2 Time Dependence Practice

Quantum Fundamentals 2023 Two electrons are placed in a magnetic field in the \(z\)-direction. The initial state of the first electron is \(\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\ i\\ \end{pmatrix}\) and the initial state of the second electron is \(\frac{1}{2}\begin{pmatrix} \sqrt{3}\\ 1\\ \end{pmatrix}\).

1. Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at \(t = 0\).
2. Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at some later time \(t\).
3. Calculate the expectation values for \(S_x\), \(S_y\), and \(S_z\) for each particle as functions of time.
4. Are there any times when all the probabilities you have calculated are the same as they were at \(t = 0\)?

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Spin Three Halves Time Dependence

Quantum Fundamentals 2023 A spin-3/2 particle initially is in the state \(|\psi(0)\rangle = |\frac{1}{2}\rangle\). This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the \(\hat{S}_x\) operator, \(\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}\)

1. Find the energy eigenvalues and energy eigenstates for the system.
2. Find \(|\psi(t)\rangle\).
3. List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
4. List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.

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Spin Fermi Estimate

Quantum Fundamentals 2023 (2 years) The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:

1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.

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Dimensional Analysis of Kets

dirac notation dimensions probability completeness relations

Completeness Relations

1. \(\left\langle {\Psi}\middle|{\Psi}\right\rangle =1\) Identify and discuss the dimensions of \(\left|{\Psi}\right\rangle \).
2. For a spin \(\frac{1}{2}\) system, \(\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1\). Identify and discuss the dimensions of \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
3. In the position basis \(\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1\). Identify and discuss the dimesions of \(\left|{x}\right\rangle \).

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Paramagnetism

Energy Temperature Paramagnetism Thermal and Statistical Physics 2020 Find the equilibrium value at temperature \(T\) of the fractional magnetization \begin{equation} \frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N} \end{equation} of a system of \(N\) spins each of magnetic moment \(m\) in a magnetic field \(B\). The spin excess is \(2s\). The energy of this system is given by \begin{align} U &= -\mu_{tot}B \end{align} where \(\mu_{tot}\) is the total magnetization. Take the entropy as the logarithm of the multiplicity \(g(N,s)\) as given in (1.35 in the text): \begin{equation} S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N} \end{equation} for \(|s|\ll N\), where \(s\) is the spin excess, which is related to the magnetization by \(\mu_{tot} = 2sm\). Hint: Show that in this approximation \begin{equation} S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N}, \end{equation} with \(S_0=k_B\log g(N,0)\). Further, show that \(\frac1{kT} = -\frac{U}{m^2B^2N}\), where \(U\) denotes \(\langle U\rangle\), the thermal average energy.

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Magnetic susceptibility

Paramagnet Magnetic susceptibility Thermal and Statistical Physics 2020

Consider a paramagnet, which is a material with \(n\) spins per unit volume each of which may each be either “up” or “down”. The spins have energy \(\pm mB\) where \(m\) is the magnetic dipole moment of a single spin, and there is no interaction between spins. The magnetization \(M\) is defined as the total magnetic moment divided by the total volume. Hint: each individual spin may be treated as a two-state system, which you have already worked with above.

1. Find the Helmholtz free energy of a paramagnetic system (assume \(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of only the ratio \(x\equiv \frac{mB}{kT}\).

2. Use the canonical ensemble (i.e. partition function and probabilities) to find an exact expression for the total magentization \(M\) (which is the total dipole moment per unit volume) and the susceptibility \begin{align} \chi\equiv\left(\frac{\partial M}{\partial B}\right)_T \end{align} as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization is \begin{align} M=nm\tanh\left(\frac{mB}{kT}\right) \end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.

3. Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the limit \(mB\ll kT\).