B.17 Chapters 37 and 38

B.17 Chapters 37 and 38

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra

Problem 1:

Show that if one takes the quantum field theory Hamiltonian operator to be

$$\hat{H}={\int }_{-\infty }^{\infty }{\hat{\Psi }}^{†}(x)\left(-\frac{1}{2m}\frac{d^2}{d{x}^2}+V(x)\right)\hat{\Psi }(x)dx$$

the field operators will satisfy the conventional Schr¨odinger equation for the case of a potential $V(x)$

Problem 2:

$A$ quantum system corresponding to indistinguishable particles interacting with each other with an interaction energy $v(x-y)$ (where $x,$ y are the positions of the particles) is given by adding a term

$$\frac{1}{2}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\hat{\Psi }}^{†}(x){\hat{\Psi }}^{†}(y)v(y-x)\hat{\Psi }(y)\hat{\Psi }(x)dxdy$$

to the free particle Hamiltonian. Just as the free-particle Hamiltonian has an expression as a momentum space integral involving products of annihilation and creation operators, can you write this interaction term as a momentum space integral involving products of annihilation and creation operators (in terms of the Fourier transform of $v(x-y)):$

Problem 3:

For non-relativistic quantum field theory of a free particle in three dimensions, show that the momentum operators $\hat{\mathbf{P}}$ (equation 38.11) and angular momentum operators $\hat{\mathbf{L}}$ (equation 38.12) satisfy the commutation relations for the Lie algebra of $E(3)$ (equations 38.13).

Problem 4:

Show that the total angular momentum operators for the non-relativistic theory of spin $\frac{1}{2}$ fermions discussed in section 38.3.3 satisfy the commutation relations for the Lie algebra of $SU(2)$

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