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Subject is physical chemistry. Normalize the two functions and show they are orthogonal.

2. Show that the first two wavefunctions of the harmonic oscillator (McQuarrie Table 5.3, p. 170) are
normalized and that they are orthogonal to each other.
a. Vo is normalized:
Y(x)
8
114 _xx ² 1/2.
e
b. ₁ is normalized:
2
3
1/4
Y₁(x) = (40³) "
Hox²) xe xx²/₂
c. o and ₁ are orthogonal:

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Follow-up Question

Can you explain how we get the circled terms please?

a) Normalization condition is √" 1460)| ² dx = 1 or [ ^ √(x) 4(x) dx =
81
de
LOHS
=
(+)
[^*^ 46(21). 46(2) dr = √ ( =) * - ** ². (=) ¹¹ é
Sa
D
1/4
L
1
e
(2) ²2500
(1
=
8
0+1
(2) ½
= (4x) x
(12²) 12.
2x²
"dx. 100
n=0 Here
2.(α)3/2
STT
So Y() is Normalized.
(16) [² 4³3 4,0x = (2-²) ³ x² = x
(²) (x) dx
1/25
42²
dx
X
42³ 1/₂ √3/₂2
=
1²
(x) 2월
2+1
(α)3/2
√T
fousing I'm² e-ox² May
=
1/2. 11/₂2
(+)
2 (0)3/2
x JTT
= 1
but √2/2 = √FT
{ using fame anda =
=0
so () is Normalized,
(c) orthogonality condition f(x) 4₁ (2) dx = 0
0.
02²
lites [ ^44) 4 (1) du = [(#)" (²) "^ 'n e
(芋)(笑)
(²) dn
14
-
81
х.е
Sating upydneo
Now
√n+1 = nim
30 11/2/2 = 1/2+1 = 1/2 √2/2 = 1/2 √
using [² f(n) da = 0 if f(x) is add
S
La
(a) n+1
Yo & 4,
to each other.
dr
odd functio
are orthogond
JL

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