# Time-independent Schrödinger equation, normalizing

• WrongMan
In summary, the problem involves an electron with energy V0 encountering a potential that is 0 for -a<x<0, V0 for 0<x<a, and infinity elsewhere. The wave function is written out as a combination of cosine and sine functions for -a<x<0 and as a linear function for 0<x<a. For normalization, continuity conditions are applied at x=0 and the norm of the wave function is set to 1. The problem can be simplified by shifting the potential by -a to create a range of [0,2a].
WrongMan

## Homework Statement

An electron coming from the left encounters/is trapped the following potential:
-a<x<0; V=0
0<x<a; V=V0
infinity elsewhere
the electron has energy V0
a)Write out the wave function
b)normalize th wave function

## The Attempt at a Solution

for -a<x<0
$$Ψ(x)=Acos(kx)+Bsin(kx)$$
$$k^2=\frac{2mV_0}{ħ^2}$$
and for 0<x<a
$$Ψ(x)=Cx+D$$
and 0 elsewhere
i used the sine and cosine because it seemed it would be better for continuity condition in x=0, if you would use exponential form please do explain why.
so this is what my teacher expects for a).
for b)
applying continuity conditions on x=0 i get:
A=D
B=C
and so:$$\int_{-a}^{0}|Ψ(x)|^2=1$$
im a bit confused here, is this the norm or the module? i think its the norm and if so ot might have been worth it to write the wave function in exponential form, so before i transcribe this big integral please clarify this for me.

Furthermore this should look like a particle traped in a box correct? i don't really understand what happens when E=V, i understand the probabiity part, it decays linearly further inside the step, correct?
And what about if E>V0 is it a particle traped in a box, but in the 0-a area the amplitude decreses? And the allowed energy levels for that area start at V0? what about penetration? and when E is smaller what happens?
Thank you!

Edit:would it be easier if i shifted the potential by -a so that it is in the range [0;2a]?

Last edited:
##B=C## isn't quite correct. You should also apply continuity conditions for ##\psi(x)## at ##x=-a## and ##x=a##.

The normalization requirement is
$$\int_{-\infty}^\infty \lvert \psi(x) \rvert^2\,dx = 1.$$ In this problem, since the wave function vanishes for ##|x|>a##, you have
$$\int_{-a}^a \lvert \psi(x) \rvert^2\,dx = 1.$$

## 1. What is the Time-independent Schrödinger equation?

The Time-independent Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a non-relativistic particle in a potential energy field.

## 2. What is the significance of the Time-independent Schrödinger equation?

The Time-independent Schrödinger equation allows us to calculate the wave function of a quantum system, which contains all the information about the system's energy and spatial distribution. This equation is essential for understanding the behavior of particles at the atomic and subatomic levels.

## 3. What is the process of normalizing the wave function?

Normalizing the wave function means rescaling it so that the total probability of finding the particle in any location is equal to 1. This allows us to interpret the wave function as a probability distribution.

## 4. Why is normalization important in the Time-independent Schrödinger equation?

Normalization is crucial in the Time-independent Schrödinger equation because it ensures that the wave function represents a physically meaningful probability distribution. It also allows us to calculate the expectation values of observables, such as position and momentum, accurately.

## 5. How is the Time-independent Schrödinger equation solved?

The Time-independent Schrödinger equation is a partial differential equation that can be solved analytically for simple systems, such as a particle in a one-dimensional box. For more complex systems, numerical methods, such as the finite difference or finite element method, are used to approximate the solution.

Replies
10
Views
794
Replies
1
Views
461
Replies
5
Views
1K
Replies
1
Views
2K
Replies
8
Views
839
Replies
2
Views
1K
Replies
2
Views
2K
Replies
4
Views
1K