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Consider electron proton scattering to be computed in the second quantized Schrodinger theory. The electrostatic interaction can be modeled as \[ V(\mathbf x_1-\mathbf x_2) = -(Ze^2)\int d\mathbf x_1\,d\mathbf x_2 \frac{\rho_e(\mathbf x_1) \rho_p(\mathbf x_2)}{|\mathbf x_1- \mathbf x_2|}\] This interaction suggests the following interaction term in the second quantized Schrodinger theory. \[ \Lsc_\text{int}=\int d\mathbf x_1d\mathbf x_2 \psi^*(\mathbf x_1)\psi(\mathbf x_1) V(\mathbf x_1-\mathbf x_2)\phi^*(\mathbf x_2)\phi(\mathbf x_2) \] where \(\psi ,\phi\) denote the electron and proton fields respectively, and \[V(\mathbf x_1-\mathbf x_2)=-\frac{(Ze^2)}{|\mathbf x_1- \mathbf x_2|}.\] Compute the electron proton scattering cross section in the center of mass frame. Use Yukawa potential \(V(\mathbf x_1-\mathbf x_2)=-V_0\dfrac{ e^{-\mu|\mathbf x_1-\mathbf x_2|}}{|\mathbf x_1- \mathbf x_2|}\) in \(\Lsc_\text{int}\). Show that in the limit \(\mu\to0\), the differential cross section coincides with the Rutherford scattering cross section.

Use the expansion of Schrodinger field \(\psi(x,t)\) in terms of plane waves \[\psi(\mathbf x,t) =\frac{1}{(2\pi)^{3/2}} \int d\mathbf k \exp(i\mathbf k.\mathbf x) a(\mathbf k,t)\] and the canonical commutation relations to prove that \[ [a(\mathbf k), a^\dagger(\mathbf k{'})] = \delta(\mathbf k-\mathbf k{'}).\]

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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 \([a(k,t), a(k{'},t) ], [a^\dagger(k,t), a^\dagger(k{'},t) ].\)