Chapter 40: Q. 26 (page 1175)

Suppose that $蠄_{1} (x)$ and $蠄_{2} (x)$ are both solutions to the Schr枚dinger equation for the same potential energy $U (x)$ . Prove that the superposition $蠄 (x) = {础蠄}_{1} (x) + {叠蠄}_{2} (x)$ is also a solution to the Schr枚dinger 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,

$- \frac{鈩�}{2 m} \frac{d^{2} 蠄 (x)}{{dx}^{2}} + U_{0} 蠄 (x) = 贰蠄 (x)$

So we can write,

$- \frac{鈩�}{2 m} \frac{d^{2} 蠄_{1} (x)}{{dx}^{2}} + U_{0} 蠄_{1} (x) = E_{1} 蠄_{1} (x) . . . . . . . . . . . . . . (1) - \frac{鈩�}{2 m} \frac{d^{2} 蠄_{2} (x)}{{dx}^{2}} + U_{0} 蠄_{2} (x) = E_{2} 蠄_{2} (x) . . . . . . . . . . . . . . . . . . (2)$

Now, add them as, $A 脳 (1) + B 脳 (2)$

Then,

$A (- \frac{鈩�}{2 m} \frac{d^{2} 蠄_{1} (x)}{{dx}^{2}} + U_{0} 蠄_{1} (x)) + B (- \frac{鈩�}{2 m} \frac{d^{2} 蠄_{2} (x)}{{dx}^{2}} + U_{0} 蠄_{2} (x)) = i 鈩� \frac{d}{dt} 蠄_{1} (x) + i 鈩� \frac{{诲蠄}_{2} (x)}{dt} - \frac{鈩�}{2 m} \frac{d^{2} ({础蠄}_{1} (x) + {叠蠄}_{2} (x))}{{dx}^{2}} + U_{0} ({础蠄}_{1} (x) + {叠蠄}_{2} (x)) = i 鈩� \frac{d ({础蠄}_{1} (x) + {叠蠄}_{2} (x)}{dt} - \frac{鈩�}{2 m} \frac{d^{2} 蠄 (x)}{{dx}^{2}} + U_{0} 蠄 (x) = i 鈩� \frac{诲蠄}{dt}$

hence proved.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Suppose that $蠄_{1} (x)$ and $蠄_{2} (x)$ are both solutions to the Schr枚dinger equation for the same potential energy $U (x)$ . Prove that the superposition $蠄 (x) = {础蠄}_{1} (x) + {叠蠄}_{2} (x)$ is also a solution to the Schr枚dinger equation.

Short Answer

Step by step solution

Given information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Nuclear Physics

Translational Dynamics

Rotational Dynamics

Measurements

Famous Physicists

Study anywhere. Anytime. Across all devices.

91影视

Suppose that 蠄1(x)and 蠄2(x)are both solutions to the Schr枚dinger equation for the same potential energy U(x). Prove that the superposition 蠄(x)=础蠄1(x)+叠蠄2(x)is also a solution to the Schr枚dinger equation.

Short Answer

Step by step solution

Given information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Nuclear Physics

Translational Dynamics

Rotational Dynamics

Measurements

Famous Physicists

Study anywhere. Anytime. Across all devices.

Suppose that $蠄_{1} (x)$ and $蠄_{2} (x)$ are both solutions to the Schr枚dinger equation for the same potential energy $U (x)$ . Prove that the superposition $蠄 (x) = {础蠄}_{1} (x) + {叠蠄}_{2} (x)$ is also a solution to the Schr枚dinger equation.