Quantum Mechanics Physics 215
Homework 4. Due in Class Friday February 25
(a) Compute and deduce the uncertainty relation: .
(b) Since , we must have . However, an eigenstate of has the property that . (Why?) Such as state therefore violates the uncertainty relation derived in part (a). Explain this apparent paradox.
(a) Taking inspiration from the algebraic solution of the harmonic oscillator, introduce creation and annihilation operators and , where j labels one of the three coordinates, x,y, or z. The position and momentum operators are defined via
where m and are arbitrary parameters. Compute the operator in terms of these creation and annihilation operators.
(b) Show, by means of a linear transformation on and , that can be written in terms of new creation and annihilation operators, and their hermitian conjugates as follows:
(c) Hence show that the eigenvalues of must be integers.
Note: You may find the discussion in Baym pp. 380-383 to be helpful.
where is positive. In the case of three dimensions. find the minimum value of which is necessary in order that there be at least one bound state. This is in contrast to the situation in one dimension where there is always binding no matter how small is. In two dimensions, is the situation analogous to the three dimensional case or to the one dimensional case?
(i) and
(ii) is anit-unitary,
i.e. ,
and , where c is a complex number.
Prove the following facts:
(a)
where depends on j but not on m. [Hint: use (i) above,
where you consider the action of on and
respectively.]
(b)
using the result of part (a) and property (ii) above.
(c) Show that by appropriate choice of in part (a), one can represent by:
where K is the (anti-unitary) complex conjugation operator.
(d) Consider an atomic system with an odd number of electrons (so that the total angular momentum of the electrons is half-integral). Show that the energy levels of the system must be at least twofold degenerate. (This is called Kramers degeneracy).