386 Quantum mechanics
together with the recursion relation forHn(y):Hn′(y) = 2nHn− 1 (y). Here, we have
used the fact thatp= ( ̄h/i)(d/dx). After some algebra, we find that
ˆaψn(x) =√
nψn− 1 (x). (9.3.29)Similarly, it can be shown that
ˆa†ψn(x) =√
n+ 1ψn+1(x). (9.3.30)These relations make it possible to bypass the eigenfunctions and work in an abstract
ket representation of the energy eigenvectors, which we denotesimply as|n〉. The
above relations can be expressed compactly as
aˆ|n〉=√
n|n− 1 〉, ˆa†|n〉=√
n+ 1|n+ 1〉. (9.3.31)Because the operator ˆa†changes an eigenvector ofHˆinto the eigenvector corresponding
to the next highest energy, it is called araising operatororcreation operator. Similarly,
the operator ˆachanges an eigenvector ofHˆinto the eigenvector corresponding to the
next lowest energy, and hence it is called alowering operatororannihilation operator.
Note that ˆa| 0 〉= 0 by definition.
The raising and lowering operators simplify calculations for the harmonic oscillator
considerably. Suppose, for example, we wish to compute the expectation value of the
operator ˆx^2 for a system prepared in one of the eigenstatesψn(x) ofHˆ. In principle,
one could work out the scary-looking integral
〈n|xˆ^2 |n〉=(
α
π 22 n(n!)^2) 1 / 2 ∫∞
−∞x^2 e−αx2
Hn^2 (√
αx)dx. (9.3.32)However, since ˆxhas a simple expression in terms of the ˆaand ˆa†,
ˆx=√
̄h
2 mω(
ˆa+ ˆa†)
, (9.3.33)
the expectation value can be evaluated in a few lines. Note that〈n|n′〉=δnn′ by
orthogonality. Thus,
〈n|ˆx^2 |n〉=̄h
2 mω〈n|(
ˆa^2 + ˆaˆa†+ ˆa†ˆa+ (ˆa†)^2)
|n〉=
̄h
2 mω[√
n(n−1)〈n|n− 2 〉+ (n+ 1)〈n|n〉+n〈n|n〉+√
(n+ 1)(n+ 2)〈n|n+ 2〉]
=
̄h
2 mω(2n+ 1). (9.3.34)Thus, by expressing ˆxand ˆpin terms of ˆaand ˆa†, we can easily calculate expectation
values and arbitrary matrix elements, such as〈n|ˆx^2 |n′〉.