Show that the three wave functions are normalized.112not normalized1orthogonal43 (x)Match the items in the left column to the appropriate blanks in the sentences and the equations on the right. Make certain each sentence and equation is complete before submitting youranswer.(VIT) ²not orthogonal√111111160√1111$1(2) = VII 01 (2) + - 02(2) + ½ 03(x)4normalized√111) = 1⁄2 01 (1) + 1/ 02(x) + VII 01 (2)- 34$2(x) -—11/01 (2) + VII 02 (2) + 1/ 03 (2)44√111VAT 01 (2) + = 02 (2) + 12 03 (2)4₁(x)√11| 45 (2) 41 (2) dx = [ (VII 01 (2) + 1702(x) + = 02 (2))*(T 01 (2) + = 02(2) + ½ $(x)) da116(VII14[ 0² (1) 01₁ (x) dx +[ 02 (2)$2(x) dx + [[03(x)øs(x) dx =All integrals involving two different ; are equal toinvolvingare equal tobecause the ; are√112 (x) = = 1/201₁ (x) + = 102(x) +4[01 (2) 01 (2) dx +- 3 (x)1√11Vs(x) = −⁄201₁ (x) + VII 02(x) + 12 (2)34[ $;(x)$1(x) dx +because the di are[ $*(x)02(x)da +Reset Help[ 02 (2) 02 (2) dx +. All integrals[03(x)03(x)dx=$; (x)03(x) dx =

Given three wave functions: ψ1x=114ϕ1x+14ϕ2x+12ϕ3xψ2x=12ϕ1x+14ϕ2x+114ϕ3xψ3x=12ϕ1x+114ϕ2x+14ϕ3x To…

Answered: Show that the three wave functions are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Show that the three wave functions are normalized.
11
2
not normalized
1
orthogonal
43 (x)
Match the items in the left column to the appropriate blanks in the sentences and the equations on the right. Make certain each sentence and equation is complete before submitting your
answer.
(VIT) ²
not orthogonal
√11
1
11
1
16
0
√11
1
1
$1(2) = VII 01 (2) + - 02(2) + ½ 03(x)
4
normalized
√11
1) = 1⁄2 01 (1) + 1/ 02(x) + VII 01 (2)
- 3
4
$2(x) -—
11/01 (2) + VII 02 (2) + 1/ 03 (2)
4
4
√11
1
VAT 01 (2) + = 02 (2) + 12 03 (2)
4
₁(x)
√11
| 45 (2) 41 (2) dx = [ (VII 01 (2) + 1702(x) + = 02 (2))*(T 01 (2) + = 02(2) + ½ $(x)) da
1
1
6
(VII
1
4
[ 0² (1) 01₁ (x) dx +
[ 02 (2)$2(x) dx + [
[03(x)øs(x) dx =
All integrals involving two different ; are equal to
involving
are equal to
because the ; are
√11
2 (x) = = 1/201₁ (x) + = 102(x) +
4
[01 (2) 01 (2) dx +
- 3 (x)
1
√11
Vs(x) = −⁄201₁ (x) + VII 02(x) + 12 (2)
3
4
[ $;(x)$1(x) dx +
because the di are
[ $*(x)02(x)da +
Reset Help
[ 02 (2) 02 (2) dx +
. All integrals
[03(x)03(x)dx=
$; (x)03(x) dx =

Expert Solution

Step 1

Given three wave functions:

$\begin{array}{rcl} ψ_{1} (x) & = & \frac{\sqrt{11}}{4} ϕ_{1} (x) + \frac{1}{4} ϕ_{2} (x) + \frac{1}{2} ϕ_{3} (x) \\ ψ_{2} (x) & = & \frac{1}{2} ϕ_{1} (x) + \frac{1}{4} ϕ_{2} (x) + \frac{\sqrt{11}}{4} ϕ_{3} (x) \\ ψ_{3} (x) & = & \frac{1}{2} ϕ_{1} (x) + \frac{\sqrt{11}}{4} ϕ_{2} (x) + \frac{1}{4} ϕ_{3} (x) \end{array}$

To show:

These waves are normalized.

Concept Used:

i) If the integral $\int ψ_{i} * (x) ψ_{i} (x) d x = 1$ , then the wave is said to be normal wave.

ii) $\int ψ_{i} * (x) ψ_{i} (x) d x = |N^{2}| \int ϕ_{i} * (x) ϕ_{i} (x) d x$ , here $ϕ (x)$ is the un-normalized function and $N$ is the normalizing constant.