Quantum - Find the formal expression of the coefficient cn(t=o)

In summary, you learned about the eigenfunctions of a Schrödinger equation and that they are normalized and have an orthogonal product.
  • #1
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Quantum -- Find the formal expression of the coefficient cn(t=o)

Hello everyone,
I'm really stuck on the first question of an exercise, so I can't start! :cry:

Homework Statement


ψ(x,t) a wave function normalized and solution of Shrodinger equation for a given potential.
I the eigenfunctions are given by the ∅n(x) (supposed to be a phi) with eigen values En so we can write ψ(x,t) as:
ψ(x,t)=Ʃ cn(t)∅n(x) = Ʃ cn(t=0)e^(-i*En*t/h)∅n(x)

a. Find the formal expression of the coefficient cn(t=o) in terms of ∅n(x), and show the maths of your result by using the orthogonality of ∅n(x).

Homework Equations





The Attempt at a Solution


Nothing, sadly.

Thanks !
 
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  • #2
Dassinia said:
Hello everyone,
I'm really stuck on the first question of an exercise, so I can't start! :cry:

Homework Statement


ψ(x,t) a wave function normalized and solution of Shrodinger equation for a given potential.
I the eigenfunctions are given by the ∅n(x) (supposed to be a phi) with eigen values En so we can write ψ(x,t) as:
ψ(x,t)=Ʃ cn(t)∅n(x) = Ʃ cn(t=0)e^(-i*En*t/h)∅n(x)

a. Find the formal expression of the coefficient cn(t=o) in terms of ∅n(x), and show the maths of your result by using the orthogonality of ∅n(x).

Homework Equations





The Attempt at a Solution


Nothing, sadly.

Thanks !

You certainly learned about the eigenfunctions of a Schrödinger equation?
What does it mean that the eigenfunctions are normal?
 
  • #3
For eigenfunctions u(x) we have H*u(x)=E*u(x)
Do you mean that the eigenfunctions are normalized ?
 
  • #4
Dassinia said:
For eigenfunctions u(x) we have H*u(x)=E*u(x)
Do you mean that the eigenfunctions are normalized ?

Not only normalized, but also... ?

Read the question:
a. Find the formal expression of the coefficient cn(t=o) in terms of ∅n(x), and show the maths of your result by using the orthogonality of ∅n(x).
What does orthogonality mean? What is the inner product of two eigenfunctions?

ehild
 
  • #5
Solved, it was really trivial.. !
Thanks for your answers !
 

1. What is the purpose of finding the formal expression of the coefficient cn(t=o) in quantum mechanics?

The formal expression of the coefficient cn(t=o) is used to determine the probability of finding a particle in a specific energy state at time t=0. This is important in understanding the behavior of quantum systems and predicting their future evolution.

2. How is the formal expression of cn(t=o) calculated?

The formal expression of cn(t=o) is calculated using the Schrödinger equation and the initial conditions of the system. It involves solving for the coefficients of the wave function using mathematical techniques such as Fourier transforms or matrix diagonalization.

3. Can the coefficient cn(t=o) be negative?

Yes, the coefficient cn(t=o) can be negative. This indicates that the probability of finding a particle in a particular energy state is negative, which may seem counterintuitive. However, in quantum mechanics, probability amplitudes can be complex numbers and the probability is given by the square of the amplitude, which can result in negative values.

4. How does the formal expression of cn(t=o) change over time?

The formal expression of cn(t=o) changes over time according to the time evolution of the quantum system. As time progresses, the coefficients cn(t=o) can become larger or smaller, indicating a change in the probability of finding a particle in a certain energy state.

5. What other factors can affect the value of cn(t=o)?

The value of cn(t=o) can be affected by factors such as the strength of the potential energy barrier, the initial conditions of the system, and the presence of other particles in the system. It can also be influenced by external factors, such as the application of a magnetic or electric field.

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