Since we have (see section 5.5 for the derivation),
〈x;X|X|ψ〉=xψ(x) 〈x;X|P|ψ〉=−i ̄h
∂
∂x
ψ(x) (5.80)
The ground state wave function is found to obey the first order differential equation,
(∂
∂x
+
Mω
̄h
x
)ψ 0 (x) = 0 (5.81)
The unique solution is given by
ψ 0 (x) =
(Mω
π ̄h
)^14exp
{−
Mω
2 ̄h
x^2
}(5.82)
where we have included the proper normalization factor.
Finally, recall that the wave functions for excited states are givenby the ground state
wave function multiplied by Hermite polynomials. This can also be recovered directly from
the operator formalism. To simplify notation, we use the scaled coordinatez=x(Mω/ ̄h)
(^12)
,
in terms of which we have
a=
1
√
2
(z+
∂
∂z
)a†=
1
√
2
(z−
∂
∂z
)(5.83)
As a result,
ψn(x) =
1
√
n! 2n
(z−
∂
∂z
)nψ 0 (z)∼Hn(x)e−
12 z^2 (5.84)
and it follows thatHn+1(z) = 2zHn(z)−Hn′(z), which is a convenient way to define the
Hermite polynomials (up to an overall normalization).
5.7 The angular momentum algebra
Both orbital angular momentum and spin satisfy the same angular momentum algebra. We
shall denote the general operators of angular momentum byJ, so that its components satisfy
the angular momentum algebra,
[Jx,Jy] =i ̄hJz [Jy,Jz] =i ̄hJx [Jz,Jx] =i ̄hJy (5.85)
A slightly more convenient notation is to use numerical subscripts for the axes, and let
J 1 =Jx,J 2 =Jy, andJ 3 =Jz, so that the algebra can be written in a single vector equation,
[Ji,Jj] =ih ̄
∑^3k=1