Chapter 40: Q. 26 (page 1175)

Suppose that $ψ_{1} (x)$ and $ψ_{2} (x)$ are both solutions to the Schrödinger equation for the same potential energy $U (x)$ . Prove that the superposition $ψ (x) = {Aψ}_{1} (x) + {Bψ}_{2} (x)$ is also a solution to the Schrödinger equation.

Short Answer

Expert verified

The prove is done.

Step by step solution

Given information

We have given, two wave function for the same potential.

We have to prove that the superposition will also be the solution.

Simplify

We know the Schrodinger equation is,

$- \frac{ℏ}{2 m} \frac{d^{2} ψ (x)}{{dx}^{2}} + U_{0} ψ (x) = Eψ (x)$

So we can write,

$- \frac{ℏ}{2 m} \frac{d^{2} ψ_{1} (x)}{{dx}^{2}} + U_{0} ψ_{1} (x) = E_{1} ψ_{1} (x) . . . . . . . . . . . . . . (1) - \frac{ℏ}{2 m} \frac{d^{2} ψ_{2} (x)}{{dx}^{2}} + U_{0} ψ_{2} (x) = E_{2} ψ_{2} (x) . . . . . . . . . . . . . . . . . . (2)$

Now, add them as, $A \times (1) + B \times (2)$

Then,

$A (- \frac{ℏ}{2 m} \frac{d^{2} ψ_{1} (x)}{{dx}^{2}} + U_{0} ψ_{1} (x)) + B (- \frac{ℏ}{2 m} \frac{d^{2} ψ_{2} (x)}{{dx}^{2}} + U_{0} ψ_{2} (x)) = i ℏ \frac{d}{dt} ψ_{1} (x) + i ℏ \frac{{dψ}_{2} (x)}{dt} - \frac{ℏ}{2 m} \frac{d^{2} ({Aψ}_{1} (x) + {Bψ}_{2} (x))}{{dx}^{2}} + U_{0} ({Aψ}_{1} (x) + {Bψ}_{2} (x)) = i ℏ \frac{d ({Aψ}_{1} (x) + {Bψ}_{2} (x)}{dt} - \frac{ℏ}{2 m} \frac{d^{2} ψ (x)}{{dx}^{2}} + U_{0} ψ (x) = i ℏ \frac{dψ}{dt}$

hence proved.

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Suppose that $ψ_{1} (x)$ and $ψ_{2} (x)$ are both solutions to the Schrödinger equation for the same potential energy $U (x)$ . Prove that the superposition $ψ (x) = {Aψ}_{1} (x) + {Bψ}_{2} (x)$ is also a solution to the Schrödinger equation.

Short Answer

Step by step solution

Given information

Simplify

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Suppose that ψ1(x)and ψ2(x)are both solutions to the Schrödinger equation for the same potential energy U(x). Prove that the superposition ψ(x)=Aψ1(x)+Bψ2(x)is also a solution to the Schrödinger equation.

Short Answer

Step by step solution

Given information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

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Suppose that $ψ_{1} (x)$ and $ψ_{2} (x)$ are both solutions to the Schrödinger equation for the same potential energy $U (x)$ . Prove that the superposition $ψ (x) = {Aψ}_{1} (x) + {Bψ}_{2} (x)$ is also a solution to the Schrödinger equation.