Unformatted text preview: PHYSICS 4455 -— QUANTUM MECHANICS
Problem Set 9 - due 11/17/2005, in class. A few more items about harmonic oscillators.
Many of you wondered in class how the raising and lowering operator formalism connects back to the explicit wave functions in coordinate space. Here, we will see how this connection is
established. An important step in the operator algebra involved the (well—founded) assumption that there
had to be a ground state, [0), with ground state energy 6,, = 1/2 in dimensionless form. When the
lowering operator at acts on this state, the result is 0: Start from the equation up) = 0 (1) (i) Express at in terms of the operators 5c and [a and write Eq. (1) in the position representation, as
a differential equation for the ground state wave function (po(x). Solve this differential equation
and normalize its solution properly. (ii) Next, we generate the ﬁrst excited state, (p 1 (x), from the ground state by using the raising
operator property ll) = 63+ 0) (2)
Write L? in terms of the operators 5c and [7 and write Eq. (2) in the position representation Find the explicit form of on (x). is it already normalized?
(iii) Now, write a general expression for (91105), by translating the relation _ l A+n
in>— ”(6010) into position space. By comparing this general form with the explicit solution for (pn(x), show
that the Hermite polynomials can be generated recursively from the relation - Hno) = We — «fire—r” The raising and lowering operator formalism is also very convenient when it comes to
computing expectation values. Here is a simple example to demonstrate its power.
(i) Starting from the explicit form of the wave function (p3(x), compute the expectation values <3m3> = [:dxcpirx)xcp3<x)
<3I22|3> = [:dxcprrchpax) by evaluating the integrals on the right hand side explicitly. (ii) Now, compute the same expectation values by expressing 5c and 5C2 in terms of d and 63+ and
exploit the properties of the raising and lowering operators, as well as the orthonormality of the
states in). If all goes well, you Will arrive at the same answer as under (i), but without doing a
single integral! . (iii) Extra credit. Exploit the operator formalism to show that the average kinetic energy and
the average potential energy are equal to each other, in eigenstate In). That is the so—called
“virial theorem”; see Liboff Problem 7.10 for parts of the solution. You should review:
Liboff, Chapter 7.1 .. 7.9, 8.5 and 8.6. ...
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- Spring '09