15.4 The Quantum Harmonic Oscillator
675
Eq. (14.2-32). If we set the constant V 0 equal to zero, the time-independent Schrödinger
equation is
Hψ −
¯
h
2
2m
d 2 ψ
dx 2 +
1
2
kx
2 ψ Eψ
(15.4-1)
We define the constants
b
2mE
¯
h
2
,
a
√
km
¯
h
(15.4-2)
so that the Schrödinger equation can be written
d
2 ψ/dx
2
+ (b − a
2 x
2 )ψ 0
(15.4-3)
This differential equation is the same as a famous equation known as the Hermite
equation (see Appendix F). Hermite solved this equation by assuming that the solution
was of the form
ψ(x) e
−ax 2 /2 S(x)
(15.4-4)
where S(x) is a power series
S(x) c 0 + c 1 x + c 2 x
2
+ c 3 x
3
+ . . .
∞
n0
c n x
n
(15.4-5)
The Hermite equation is named for
Charles Hermite, 1822–1901, a great
French mathematician who made many
contributions to mathematics, including
the proof that e (2.71828. . .) is a
transcendental irrational number.
with constant coefficients c 1 , c 2 , c 3 , . . . When Eqs. (15.4-4) and (15.4-5) are substituted
into the Hermite equation and the exponential factor is canceled, an equation with power
series on both sides results. If two power series are equal to each other for all values
of the independent variable, the coefficients of the same power in the two series must
be equal to each other. From this fact Hermite obtained the relation
c n+2
2an + a − b
(n + 2)(n + 1)
c n (n 0, 1, 2, . . .)
(15.4-6)
Exercise 15.9
Assume that the following equation is valid for any value of x:
a 0 + a 1 x + a 2 x 2 + . . . b 0 + b 1 x + b 2 x 2 + . . .
Show that a 0 b 0 by letting x 0. Differentiate both sides of the equation with respect to x,
and show that a 1 b 1 by letting x 0 in the formula for the derivative. Continue to show that
a n b n for any value of n.
Equation (15.4-6) is called a recursion relation. Given a value of c n , it provides a value
for c n+2 . If we pick any value for c 0 and any other value for c 1 and let the recursion
675
Eq. (14.2-32). If we set the constant V 0 equal to zero, the time-independent Schrödinger
equation is
Hψ −
¯
h
2
2m
d 2 ψ
dx 2 +
1
2
kx
2 ψ Eψ
(15.4-1)
We define the constants
b
2mE
¯
h
2
,
a
√
km
¯
h
(15.4-2)
so that the Schrödinger equation can be written
d
2 ψ/dx
2
+ (b − a
2 x
2 )ψ 0
(15.4-3)
This differential equation is the same as a famous equation known as the Hermite
equation (see Appendix F). Hermite solved this equation by assuming that the solution
was of the form
ψ(x) e
−ax 2 /2 S(x)
(15.4-4)
where S(x) is a power series
S(x) c 0 + c 1 x + c 2 x
2
+ c 3 x
3
+ . . .
∞
n0
c n x
n
(15.4-5)
The Hermite equation is named for
Charles Hermite, 1822–1901, a great
French mathematician who made many
contributions to mathematics, including
the proof that e (2.71828. . .) is a
transcendental irrational number.
with constant coefficients c 1 , c 2 , c 3 , . . . When Eqs. (15.4-4) and (15.4-5) are substituted
into the Hermite equation and the exponential factor is canceled, an equation with power
series on both sides results. If two power series are equal to each other for all values
of the independent variable, the coefficients of the same power in the two series must
be equal to each other. From this fact Hermite obtained the relation
c n+2
2an + a − b
(n + 2)(n + 1)
c n (n 0, 1, 2, . . .)
(15.4-6)
Exercise 15.9
Assume that the following equation is valid for any value of x:
a 0 + a 1 x + a 2 x 2 + . . . b 0 + b 1 x + b 2 x 2 + . . .
Show that a 0 b 0 by letting x 0. Differentiate both sides of the equation with respect to x,
and show that a 1 b 1 by letting x 0 in the formula for the derivative. Continue to show that
a n b n for any value of n.
Equation (15.4-6) is called a recursion relation. Given a value of c n , it provides a value
for c n+2 . If we pick any value for c 0 and any other value for c 1 and let the recursion