Wavefunction in the energy representation

In summary, the conversation discusses finding the 'wavefunction in the energy representation' for a given equation involving normalized energy eigenfunctions. The solutions proposed involve using completeness of states to represent the wavefunction as a sum of coefficients at different energies, with the probabilities of finding a specific energy given by the squared coefficients. The use of Dirac and Kronecker deltas is also mentioned as a possible approach. Further clarification is needed to determine the exact solution.
  • #1
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Homework Statement



[tex]\psi(x)=\frac{3}{5}\chi_{1}(x)+\frac{4}{5}\chi_{3}(x)[/tex]

Both [tex]\chi_{1}(x) \chi_{3}(x)[/tex] are normalized energy eigenfunctions of the ground and second excited states respectivley. I need to find the 'wavefunction in the energy representation'


The Attempt at a Solution



I can find the expectation value of the energy but what is the wavefunction in the energy representation?
 
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  • #2
It looks like it is given in the energy representation already. Except I'd write it as:

[tex]\left|\psi\right> = \frac{3}{5}\left|E_1\right>+\frac{4}{5}\left|E_3\right>[/tex]
 
  • #3
I'm not so sure. I think what they are after is [tex]\psi(E)[/tex] i.e. a function of energy. I suppose in analogy with [tex]\psi(x) = \left<x|\psi\right>[/tex] it would be [tex]\psi(E) = \left<E|\psi\right>[/tex]. Completeness of states could be of help. I suspect you should get delta functions.
 
  • #4
phsopher said:
I'm not so sure. I think what they are after is [tex]\psi(E)[/tex] i.e. a function of energy. I suppose in analogy with [tex]\psi(x) = \left<x|\psi\right>[/tex] it would be [tex]\psi(E) = \left<E|\psi\right>[/tex]. Completeness of states could be of help. I suspect you should get delta functions.
I wouldn't have thought of that, but yeah, that might be it.
 
  • #5
My guess at what they want: (I think it's the same as what Phsopher said, but possibly a bit more intuitive)

By completeness, you can represent any wave-function:

[tex]\Psi (x) = \sum_{n=1}^{\infty} c_n\psi_n(x)[/tex]

Where the [tex]\psi_n[/tex] represent energy eigenfunctions. The energy representation of the wave function would probably then just be the coefficients [tex]c_n[/tex] at different energies. The probabilities of finding a specific energy would be [tex]|c_n|^2[/tex]. This is a bit weird, because it's not any real function that I can think of (dirac deltas can't really be squared properly...as far as I know...so I don't think ).

I'm sort of stumped on this one too!

EDIT:

Oh, I just thought, maybe instead of using Dirac Deltas, you can use Kronecker deltas...

Perhaps:

[tex]\Psi(E) = \frac{3}{5}\delta_{E, E_1} + \frac{4}{5}\delta_{E, E_3}[/tex]

Not 100% on this though.
 
Last edited:

1. What is a wavefunction in the energy representation?

A wavefunction in the energy representation is a mathematical function that describes the probability of finding a particle at a certain energy level. It is used to understand the behavior of quantum systems and is an essential concept in quantum mechanics.

2. How is the wavefunction related to energy?

The wavefunction represents the energy states of a quantum system. It contains all the information about the energy levels that a particle can occupy, as well as the probability of finding the particle at each energy level.

3. What is the difference between position and energy representations of a wavefunction?

In the position representation, the wavefunction describes the probability of finding a particle at a given position in space. In the energy representation, the wavefunction describes the probability of finding a particle at a given energy level. The two representations are related by a mathematical transformation known as a Fourier transform.

4. How is the wavefunction used in quantum mechanics?

The wavefunction is a fundamental concept in quantum mechanics and is used to describe the behavior of particles at the subatomic level. It is used to calculate the probability of a particle being in a certain energy state and is essential for understanding phenomena such as quantum tunneling and superposition.

5. Can the wavefunction be observed or measured?

No, the wavefunction is a mathematical concept and cannot be directly observed or measured. However, the predictions made by the wavefunction can be tested and verified through experiments, providing evidence for its validity in describing the behavior of quantum systems.

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