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If two wave functions ψ1 (x,t), ψ2 (x,t) are solutions to the (one dimensional) time dependent Schroedinger eqn. show that ψ = Aψ1 + Bψ2 is also a solution, A and B are complex constants.

If two wave functions ψ1 (x,t), ψ2 (x,t) are solutions to the (one dimensional) time dependent Schroedinger eqn. show that ψ = Aψ1 + Bψ2 is also a solution, A and B are complex constants.

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Subject: Physics - Jr/Senior level Quantum Mechanics - 

If two wave functions ψ1 (x,t), ψ2 (x,t) are solutions to the (one dimensional) time dependent Schroedinger eqn. show that ψ = Aψ1 + Bψ2 is also a solution, A and B are complex constants.  I started by plugging Aψ1 + Bψ2 into the time dependent Schroedinger equation but not sure where to go from there.  Thank you! 

Expert Solution
Step 1

Given,

ψ1x,t and ψ2x,t are the solutions of time-dependent Schrodinger's equation.

Then,

-h22m∂2ψ1x,t∂x2+Vx,tψ1x,t=-ih∂ψ1x,t∂t.....................1

And

-h22m∂2ψ2x,t∂x2+Vx,tψ2x,t=-ih∂ψ2x,t∂t.....................2

 

Now we have to prove that the wave function ψ1x,t+ψ2x,t is also a solution of Schrodinger's equation.

 

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