How Do You Write a Superposition Wavefunction for an Electron in a 1D Box?

[coconutfreak]

1. PROBLEM

Q: write down an expression for a superposition wavefunction [itex]\Psi[/itex](x) for an electron in a 1D box of length L consisting of the n = 1, 2, and 3 states. show that C₁² + C₂² + C₃² = 1, and C_n represents the coefficients of the n state.

2. RELEVANT EQUATIONS

see word document. thanks.

3. MY ATTEMPT

Am I supposed to integrate or differentiate the equations at all?
I don't know how to even start this Q.



Thanks a lot for your help :)

 

[ideasrule]

It looks like you know the equation for the nth state. A "superposition" simply means the wavefunction is in the form c1*Ψ₁(x) + c2*Ψ₂(x) + c3*₃Ψ(x), where c1, c2, and c3 are complex numbers.

To start on the question, write out the wavefunction and take its inner product with itself. Remember that the eigenfunctions are orthogonal. What do you get?

 

[coconutfreak]

i don't understand..
i am so sorry..

i know how to get Ψ1(x), Ψ2(x) and Ψ3(x) by simply substituting n = 1, 2, or 3 into the wavefunction equation.

what do you mean by inner product? dot product?

but i don't know how to continue from here.

thanks a lot for your help.

 

[ideasrule]

i don't understand..
i know how to get Ψ1(x), Ψ2(x) and Ψ3(x) by simply substituting n = 1, 2, or 3 into the wavefunction equation.

Yup!

what do you mean by inner product? dot product?

but i don't know how to continue from here.

thanks a lot for your help.

Hmm, I don't know what you've already learned, so it's kind of hard to explain. Have you seen the notation <a|b> before? That's the inner product of two functions, and it's defined as the integral of the a*b (complex conjugate of a multiplied by b).

If you haven't seen it before, no worries. Just normalize the wavefunction in the way that you usually normalize wavefunctions. However, make sure to normalize the entire wavefunction, not just the individual states.

 

[rwisz]

I would be happy to provide a response to this content. First, let's define what a superposition wavefunction is. A superposition wavefunction is a mathematical representation of a quantum system that is in a combination of multiple states. In this case, we are considering an electron in a 1D box with three possible states (n=1, 2, and 3). The wavefunction, denoted by Ψ(x), is a complex-valued function that describes the probability amplitude of the electron at a given position x in the box.

To answer the problem, we need to write down an expression for Ψ(x) that includes all three states. One way to do this is to use the linear combination of states, where the coefficients Cn represent the contribution of each state to the overall wavefunction. This can be written as:

Ψ(x) = C1Ψ1(x) + C2Ψ2(x) + C3Ψ3(x)

Where Ψn(x) is the wavefunction for the nth state, which can be determined using the relevant equations provided in the word document.

To show that the coefficients Cn satisfy the condition C12 + C22 + C32 = 1, we need to use the normalization condition. This means that the total probability of finding the electron in the box must be equal to 1. We can express this as:

∫Ψ(x)*Ψ(x) dx = 1

Where Ψ(x)* is the complex conjugate of Ψ(x). Substituting our expression for Ψ(x) into this equation, we get:

∫(C1Ψ1(x) + C2Ψ2(x) + C3Ψ3(x))*(C1Ψ1(x) + C2Ψ2(x) + C3Ψ3(x)) dx = 1

Expanding and simplifying, we get:

C12∫Ψ1(x)*Ψ1(x) dx + C22∫Ψ2(x)*Ψ2(x) dx + C32∫Ψ3(x)*Ψ3(x) dx + 2C1C2∫Ψ1(x)*Ψ2(x) dx + 2C1C3∫Ψ1(x)*Ψ3(x) dx + 2C2C3∫Ψ2(x