# Quantum mechanics - infinite square well problem

• Graham87
In summary, the conversation revolved around solving the integral in part d) of a problem. The person had successfully solved part c) but was struggling with the integral in part d). They considered rewriting part c) into an initial wave function with constants, but were unsure how to proceed. Eventually, they were able to relate the integral to one of the eigenfunctions and use orthonormality to solve it. It was suggested to use general properties of eigenfunctions as much as possible.
Graham87
Homework Statement
For a wave function evaluate the following integral
Relevant Equations
Infinite square well equations

I have solved c), but don’t know how to solve the integral in d.
It looks like an integral to get c_n (photo below), but I still can’t figure out what to make of c) in the integral of d).

I also thought maybe you can rewrite c) into an initial wave function (photo below) with A,x,a but don’t know how.

Thanks!

Delta2
Graham87 said:
I have solved c), but don’t know how to solve the integral in d.

Consider using a trig identity to write ##\sin\left(\frac{2\pi}{a}x\right)\cos\left(\frac{2\pi}{a}x\right)## in a way that will be helpful in evaluating the integral.

Graham87 and Delta2
I tried like this already, but I still don’t know how to deal with c) in the integral.

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Graham87 said:
I tried like this already, but I still don’t know how to deal with c) in the integral.
OK, good. Can you relate ##\large\frac{\sin\left(\frac{4\pi x}{a}\right)}{2}## to one of the ##\psi_n(x)##?

Graham87 and Delta2
TSny said:
OK, good. Can you relate ##\large\frac{\sin\left(\frac{4\pi x}{a}\right)}{2}## to one of the ##\psi_n(x)##?
ψ4(x) ?

Delta2
Graham87 said:
ψ4(x) ?
Exactly what is the relation between ##\large\frac{\sin\left(\frac{4\pi x}{a}\right)}{2}## and ##\psi_4(x)##?

Graham87 and Delta2
TSny said:
Exactly what is the relation between ##\large\frac{\sin\left(\frac{4\pi x}{a}\right)}{2}## and ##\psi_4(x)##?
Same n?
How does this look?

I don’t think it’s correct because c_4 is not the same answer.

Graham87 said:
How does this look?View attachment 303556
That will work.

However, it's maybe a little nicer to show ##\large\frac{\sin\left(\frac{4\pi x}{a}\right)}{2} = \frac{\sqrt{2a}}{4}\psi_4(x)## . Then you don't need to do any explicit integration. Just use the orthonormality of the ##\psi_n(x)##.

Delta2, Graham87, PeroK and 1 other person
TSny said:
That will work.

However, it's maybe a little nicer to show ##\large\frac{\sin\left(\frac{4\pi x}{a}\right)}{2} = \frac{\sqrt{2a}}{4}\psi_4(x)## . Then you don't need to do any explicit integration. Just use the orthonormality of the ##\psi_n(x)##.
So like this?

I found the integral d) similar to this.

Is it wrong to assume that c_4 should be the same as my answer in d)?
In that case my answer in d is not correct.

$$\frac{\sqrt{2a}}{4} \int \psi_4^2\,dx$$ and $$\int \psi_4^2\,dx$$ equal? I think you should be able to answer that on your own. :)

Graham87
You need to calculate ##\int_0^a{\sin\left(\frac{2\pi x}{a}\right)\cos\left(\frac{2\pi x}{a}\right)}\Psi(x,0)dx##

Write this as ##\frac{\sqrt{2a}}{4}\int_0^a\psi_4(x)\Psi(x,0)dx##.

Substitute the given expression for ##\Psi(x,0)## and evaluate using the orthonormality of the ##\psi_n(x)##.

You should get the same answer as you got in post #7.

Delta2, PeroK and Graham87
Big thanks! Got it!

TSny
Graham87 said:
Big thanks! Got it!
In general, it's a good habit to use the notation ##\psi_n(x)## and use the general properties of eigenfunctions - especially orthonormality - as much as possible. And only resort to the specific eigenfunctions when necessary.

Graham87, Delta2 and TSny

## 1. What is the infinite square well problem in quantum mechanics?

The infinite square well problem is a theoretical scenario in quantum mechanics where a particle is confined to a one-dimensional box with infinitely high potential walls on either side. This problem is used to understand the behavior of particles in confined spaces and to demonstrate the principles of quantum mechanics.

## 2. How is the infinite square well problem solved?

The infinite square well problem is solved by using the Schrödinger equation, a fundamental equation in quantum mechanics that describes the behavior of particles in terms of wave functions. By solving the Schrödinger equation for the infinite square well potential, we can determine the allowed energy levels and corresponding wave functions for the particle.

## 3. What are the implications of the infinite square well problem?

The infinite square well problem has several important implications in quantum mechanics. It shows that particles can only exist in discrete energy states, rather than having a continuous range of energies. It also demonstrates that the position and momentum of a particle cannot be known simultaneously with certainty, a fundamental principle known as the Heisenberg uncertainty principle.

## 4. How does the width of the well affect the energy levels in the infinite square well problem?

In the infinite square well problem, the width of the well affects the energy levels in a quantized manner. As the width of the well decreases, the energy levels become more closely spaced, meaning that the particle has a higher probability of being found in a particular location within the well.

## 5. What real-world applications does the infinite square well problem have?

The infinite square well problem has many real-world applications, particularly in the field of nanotechnology. It helps us understand the behavior of electrons in quantum dots, which are tiny structures used in electronic devices. It also has applications in understanding the properties of atoms and molecules, and in developing new materials with unique quantum properties.

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