A study of classical waves tells us that a standing wave can be expressed as a sum of two traveling waves, Quantum- mechanical traveling waves, are of the form Psi(x,t)=Ae^(i(kx−omega t)) Show that the infinite well's standing-wave function can be expressed as a sum of two traveling waves.

garkochenvz

garkochenvz

Answered question

2022-08-11

A study of classical waves tells us that a standing wave can be expressed as a sum of two traveling waves, Quantum- mechanical traveling waves, are of the form Ψ ( x , t ) = A e i ( k x − ω t ) Show that the infinite well's standing-wave function can be expressed as a sum of two traveling waves.

Answer & Explanation

Pasrbekwp

Pasrbekwp

Beginner2022-08-12Added 10 answers

Given information:
Ψ ( x , t ) = A e i ( k x − ω t ) 
Approach:
We add two waves that are going in opposite directions and at different velocities in order to demonstrate that the standing wave function of an infinite well may be written as the sum of two waves. As a result, one of the waves has a negative sign for k. We get:
Ψ 1 ( x , t ) = A e i ( k x − ω t ) 
Ψ 2 ( x , t ) = − A e i ( − k x − ω t ) 
Ψ T ( x , t ) = Ψ 1 + Ψ 2 
= A e i ( k x − ω t ) − A e i ( − k x − ω t ) 
= A ( e i ( k x − ω t ) − e i ( − k x − ω t ) )
= A e − i ω t ( e i k x − e − i k x )
= 2 A i ( e i k x − e − i k x 2 i ) e − i ω t 
We know that: e i k x − e − i k x 2 i = sin ⁡ ( k x )
So, we can prove that Ψ ( x , t ) = A sin ⁡ ( k x ) e − i ω t 

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