Superposition representation of particle state in 1-d infitne well (SUPERPOSITION?)

In summary, the question is asking for steps to express a given function as a superposition in the basis of solutions of the time independent Schrodinger equation for a particle in a 1-D infinite potential well. This involves using trigonometric identities to express the given function as a sum of functions of the form sin(nπx/a), and then normalizing the first given function.
  • #1
mak015
1
0

Homework Statement


Here it is: a particle in 1-d infinite potential well starts in state [tex]\Psi[/tex](x,0) = A Sin[tex]^{3}[/tex]([tex]\pi[/tex]*x/a): 0[tex]\leq[/tex]x[tex]\leq[/tex]a.

Express [tex]\Psi[/tex](x,0) as a superposition in the basis of the solutions of the time independent schrodinger eq for this system, [tex]\phi_{n}[/tex](x) = (2/a)[tex]^{1/2}[/tex] Sin(n*[tex]\pi[/tex]* x /a).

Homework Equations


What are the steps to take to bring me to the correct answer. I'm not sure what exactly the question is asking for, or rather how to show it.


The Attempt at a Solution


I assume to know that superposition states that [tex]\Psi[/tex](x,0) = [tex]\sum[/tex][tex]C_{n}[/tex]*[tex]\phi_{n}[/tex](x).

Then since they are bound (therefore orthogonal) it can be said that
[tex]\int\Psi(x,0)\Psi^{*}_{m}(x,0)dx[/tex] = 1 from 0 to a.

Can it then be said that [tex]\sum[/tex]C[tex]_{n}[/tex][tex]\int\phi_{n}(x)\Psi^{*}_{m}(x,0)dx [/tex]also equals 1, therefor equaling the above eqn?

From here, I don't know how to approach the goal of this problem.


More parts to the question ask for solving for [tex]C_{n}[/tex] and normalizing the first given fctn.


Any help is greatly appreciated!

Mark
 
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  • #2


Welcome to PF.

Not 100% sure, but I'm thinking the idea is to use trig identities to express sin3 in terms of sin(π x/a), sin(2π x/a), etc.

Haven't worked this through to know for sure that will work though.
 
  • #3


Okay, I've looked at this one some more.

A helpful identity is

sinθ = (eiθ - e-iθ) / (2i)​

So sin3θ = ?
 

1) What is the superposition representation of particle state in 1-d infinite well?

The superposition representation of particle state in 1-d infinite well is a mathematical model that describes the probability of finding a particle in a particular position within the well. It takes into account the concept of quantum superposition, where a particle can exist in multiple states simultaneously.

2) How is the superposition representation used in quantum mechanics?

The superposition representation is a fundamental concept in quantum mechanics, as it allows us to understand the behavior of particles at the atomic and subatomic levels. It is used to calculate the probability of finding a particle in a certain state, and to predict the outcomes of experiments in quantum systems.

3) Can you explain the mathematical equation for the superposition representation?

The mathematical equation for the superposition representation is known as the wave function, represented by the symbol Ψ (psi). It is a complex-valued function that describes the state of a quantum system at a certain time. The square of the wave function represents the probability density of finding a particle in a particular state.

4) What is the significance of the 1-d infinite well in quantum mechanics?

The 1-d infinite well is a simplified model used to understand the behavior of particles in a confined space. It is used to demonstrate the principles of quantum mechanics, such as the superposition of states and the quantization of energy levels. It also serves as a building block for more complex quantum systems.

5) How does the superposition representation relate to the uncertainty principle?

The superposition representation is closely related to the uncertainty principle, which states that it is impossible to know both the exact position and momentum of a particle simultaneously. This is because the superposition of multiple states means that the particle does not have a definite position, and therefore its momentum cannot be precisely determined.

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