Quantum field theory

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 * Show that the scalar (or inner) product of a quantum state \(\phi\) with itself can be written as an integral
 * Show that the scalar (or inner) product of two different quantum states \(\phi\) and \(\psi\) and the operator Â can be written as an integral
 * Show that the scalar (or inner) product of two different quantum states \(\phi\) and \(\psi\) and the position operator \(x\) can be written as an integral
 * Show that the scalar (or inner) product of position \(x\) and momentum \(p\) can be written as an exponential in terms of \(x\) and \(p\)
 * Show that the scalar (or inner) product of basis state \(x\) and quantum state \(\phi\) with the momentum operator \(p\) can be written as a derivative
 * Show that the scalar (or inner) product of basis state \(p\) and quantum state \(\phi\) with the position operator \(x\) can be written as a derivative
 * Show that the scalar (or inner) product of basis state \(p\) and quantum state \(\phi\) with a function of the position operator \(V(x)\) can be written as an integral
 * Prove and demonstrate the existence of ladder operators for the simple harmonic oscillator
 * Show that the ladder operator commutator is consistent with the momentum and position commutator
 * Show that number states are eigenstates of the Hamiltonian operator for a simple harmonic oscillator
 * Prove that the raising operator raises the state of the simple harmonic oscillator
 * Prove that the lowering operator lowers the state of the simple harmonic oscillator
 * Write an arbitrary number state as an excitation of the ground state for the simple harmonic oscillator
 * Calculate expectation values for odd powers of \(x\) in state \(n\) using ladder operators
 * Calculate expectation values for \(x^2\) in state \(n\) using ladder operators
 * Calculate expectation values for \(x^4\) in state \(n\) using ladder operators
 * Derive the equations of motion for a simple harmonic oscillator
 * Determine the momentum of a simple harmonic oscillator
 * Quantise the coupled harmonic oscillator
 * Derive the equations of motion for arbitrarily many coupled harmonic oscillators with alternating masses
 * Use the Euler-Lagrange equation
 * Show that the Hamiltonian of the quantum elastic chain can be represented as a sum of harmonic oscillator Hamiltonians
 * Show that the Klein-Gordon field satisfies the continuity equation
 * Derive the quantised Hamiltonian corresponding to the uncharged Klein-Gordon equation
 * Derive the quantised Hamiltonian corresponding to the charged Klein-Gordon equation
 * Show that the Dirac field satisfies the continuity equation
 * Derive the quantised Hamiltonian corresponding to the Dirac equation
 * Show that if the raising operator acts on an eigenstate of the number operator, the result is also an eigenstate of the number operator
 * Show that if the lowering operator acts on a non-zero eigenstate of the number operator, the result is also an eigenstate of the number operator
 * Represent an arbitrary number state \(n\) in terms of creation operators and the vacuum state
 * Second quantise the one-body kinetic energy operator
 * Find the two-body potential energy
 * Show that the many-body wavefunction for bosons is symmetric
 * Show that the many-body wavefunction for fermions is antisymmetric
 * Write the Fock-space state in terms of creation operators acting on the vacuum state
 * Diagonalise the cubic lattice tight-binding Hamiltonian
 * Find the Hamiltonian for the interaction between phonons and optical phonons
 * Show that the bosonic commutation relations of the creation and annihilation operators are preserved by the Bogoliubov transformation
 * Act on the Hamiltonian \(H=F(a^\dagger a+b^\dagger b)+G(a^\dagger b^\dagger+ab)\) with the Bogoliubov transformation
 * Find the excitation spectrum for a weakly interacting bosonic superfluid
 * Find the fraction of bosons not in the concentrate in a weakly interacting bosonic superfluid
 * Show that the Holstein-Primakoff transformation satisfies the spin commutation relations
 * Show that the Schwinger representation operators satisfy the spin commutation relations
 * Find the eigenstate of the total spin operator in the Schwinger boson representation
 * Find the energy levels of a system of two interacting spins
 * Use the Holstein-Primakoff transformations to find the low-lying excitation energies for a system of two interacting spins
 * Find the quadratic bosonic theory that describes a 1-dimensional Heisenberg ferromagnet
 * Use the mean field approximation to simplify the BCS superconductivity Hamiltonian
 * Use the Bogoliubov transformation to diagonalise the BCS superconductivity Hamiltonian
 * Find the ground state of the BCS superconductor
 * Find the gap energy \(\Delta\) of the BCS superconductor
 * Derive the propagator for a free particle from first principles
 * Derive the propagator for a free particle from the Feynman path integral
 * Express the wavefunction \(\phi(q,t)\) in terms of an integral of the initial state \(\phi(q,0)\)
 * Find the probability for finding a free particle within \(dq\) of a point \(q\)
 * Find the bosonic partition function
 * Use Grassman integration

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