Browse & Filter

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• The Lagrangian density for the Schrodinger equation is given to be \[\Lsc = i\hbar\psi^*(x,t)\pp[\psi(x,t)]{t} - \frac{\hbar^2}{2m} |\nabla \psi|^2 - \psi^*(x,t)V(x)\psi(x,t)\] Verify that the Euler Lagrange equations for the Schrodinger field coincide with the Schrodinger equation.
• Find the Hamiltonian of the system. Use Poisson brackets to obtain the Hamiltonian equations of motion.
• Verify that the Hamilton's equations imply the Euler Lagrange equation of motion.

Taking the case of free Schrodinger field answer the following questions.

• Find Heisenberg equations of motion for the operators \(a(k,t)\) and \(a^\dagger(k,t)\)
• Solve the equations of motion and expre
• \(a(k,t)\) and \(a^\dagger(k,t)\) as functions of time. Calculate the unequal time commutators \[\big[a(k,t), a(k{'}, t{'})\big],\quad \big[a^\dagger(k,t), a^\dagger(k{'}, t{'})\big],\quad \big[a(k,t), a^\dagger(k{'}, t{'})\big].\]
• Use your answers and work out the unequal time commutator \[\big[\psi(x,t),\psi^\dagger(x{'},t{'})\big].\]
• Use your result for unequal time commutator and express \(\psi(x_1,t_1) \psi^\dagger(x_2,t_2)\) in a normal ordered form.

Define \(a(k,t)\) and \(a^\dagger(k,t)\) as Fourier coefficients of free field \(\psi(x,t)\). \[ \psi(x,t)= \frac{1}{(2\pi)^3}\int dx e^{ikx} a(k);\psi^\dagger(x,t)= \frac{1}{(2\pi)^3}\int dx e^{-ikx} a^\dagger(k); \]

• Use ETCR to show that \[ [(a(k,t), a^\dagger(k{'},t)]= \delta(k-k{'}).\] What are the values of equal time commutators \([a(k,t), a(k{'},t) ], [a^\dagger(k,t), a^\dagger(k{'},t) ].\)

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Consider free Schrodinger equation as a quantized field.

• Show that \begin{equation} G(x -x{'}, t -t{'} ) = \matrixelement{0}{T ψ\psi(x, t)\psi^\dagger (x{'}, t{'}))}{0} \end{equation} obeys the equation for Green function of the free particle Schrodinger equation.
• Use expansion of the field operators in terms of free particle wave function \(N \exp(ikx - iE_k t)\), where \(E_k = \frac{\hbar^2k^2}{2m}\). Obtain an explicit expression for this time ordered product as a function of \(x, t, x{'} , t{'}\).
• Have you seen this object before? Where?

Write the transformation properties of free Dirac field of under a Lorentz boost by velocity \(vec{v}=v(0,0,1)\). Show that the Dirac wave function for a particle with momentum \(\vec{p}=p(0,0,1)\) and mass \(m\) can be obtained by applying an appropriate boost on wave function for a particle at rest. Demonstrate this only for positive energy solution with spin along \(z\)-axis

Let \(n^\mu\) be a spacelike four vector satisfying \(n^\mu n_\mu=-1\), show that

• the eigenvalues of \(\gamma_5n\!\!\!/ \) are \(\pm1\).
• \((\gamma_5 n\!\!\!/)^2 =1\)
• If \(p^\mu n_\mu=0\), \(p\!\!\!/\) commutes with \(\gamma_5 n\!\!\!/ \)
• Free particle solutions \(u(p)\) and \(v(p)\) of Dirac equation \[ (p\!\!\!/ -M )u(p)=0, \qquad (p\!\!\!/  + M)v(p)=0\] can also be taken to be eigenvectors of \(\gamma_5 n\!\!\!/   \)  

Does there exist an invertible matrix \(S\) such that \[ S \gamma_\mu S^{-1} = \gamma_\mu'\] where \[\gamma_1'= \gamma_2\gamma_3, \quad \gamma_2'=\gamma_3\gamma_1, \quad \gamma_3'= \gamma_1\gamma_2, \gamma_4'=\gamma_5\gamma_4?\]

Let \(p^\mu\) be a time like momentum vector. Let \(n^\mu\) be a four vector such that \begin{equation} n^\mu p_\mu=0, \text{ and } n^\mu n_\mu=-1. \end{equation} How many such independent four vectors \(n^\mu\) exist? How that the operators \begin{equation} \Pi^\pm_n =\frac{1}{2}\big(1\pm \gamma_5 n\!\!\!{n}\big) \end{equation} are projection operators satisfying. \begin{equation} \Pi^{(+)2}_n = \Pi^{(-)2}_n = I, \qquad \Pi^{(+)}_n \Pi^{(-)}_n =0. \end{equation} The notation here is same as in Bjorken and Drell, Gasiorowicz.

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Compute the matrix element \[ \matrixelement{s,\vec{q}}{J_\mu(x)}{r,\vec{p}}\] % where \(J_\mu(x)\) is the current for a Dirac particle and \( \ket{s,\vec{q}}\) and \(\ket{r,\vec{p}} \) denote one particle states of spin half particle with spin and momenta as specified.