Dosen't energy eigenvalue depend on x?

In summary, the conversation discusses the relationship between potential energy and energy eigenvalues in a quantum system. It is explained that the total energy can be calculated using the wave function and the Hamiltonian operator, and that the potential energy can be calculated using only the wave function. The question is raised whether the energy eigenvalue remains the same throughout the region beyond x=a, and it is suggested that this is guaranteed since the state would not be an energy eigenfunction otherwise.
  • #1
rar0308
56
0
there is potential V(x).
If at some point x=a wavefunction have some energy eigenvalue,
then Is it guaranteed that It has same energy throughout whole region?
Where can I find explanation about this?
 
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  • #2
rar0308 said:
there is potential V(x).
If at some point x=a wavefunction have some energy eigenvalue,
then Is it guaranteed that It has same energy throughout whole region?
Where can I find explanation about this?

You can't talk about the energy at a point. You get the energy as an expectation value (not eigenvalue, unless you are in an eigenstate of the Hamiltonian). So, if you want to know the total energy of the system described by the wave function ##\psi(x)##, you calculate
$$
E = \int_{-\infty}^{\infty} \psi^*(x) \hat{H} \psi(x) dx.
$$
If you want only the potential energy, you use instead
$$
E_\mathrm{pot} = \int_{-\infty}^{\infty} \psi^*(x) V(x) \psi(x) dx.
$$
(I'm assuming that the wave function is normalized.)
 
  • #3
Let quantum state be given as one of energy eigenfunctions.
V(x) is unknown except some point x=a.
Calculated energy eigenvalue using Hamiltonian operator at x=a.
Is it guaranteed that energy eigenvalue is the same throughout whole region other than x=a?
 
  • #4
Yes, otherwise the state wouldn't be an energy eigenfunction would it?
 
  • #5
dauto is Right. How can i delete this thread?
 

1. Why doesn't the energy eigenvalue depend on x?

The energy eigenvalue is a property of a quantum mechanical system and represents the total energy of a particle in that system. It is independent of the position of the particle, as it is determined by the overall potential energy of the system and not by the position of the particle within it.

2. How is the energy eigenvalue determined?

The energy eigenvalue is determined by solving the Schrödinger equation, which describes the time evolution of a quantum system. This equation takes into account the potential energy of the system and the wave function of the particle, and the solutions to this equation yield the energy eigenvalues for the system.

3. Can the energy eigenvalue change?

The energy eigenvalue is a constant value for a given quantum system, but it can change if the system is perturbed or if the potential energy of the system changes. In these cases, the solutions to the Schrödinger equation will yield different energy eigenvalues.

4. How does the energy eigenvalue relate to the energy of the particle?

The energy eigenvalue represents the total energy of a particle in a quantum system. However, it is not the same as the kinetic energy or potential energy of the particle, as these quantities are dependent on the position and momentum of the particle. The energy eigenvalue takes into account all forms of energy in the system.

5. Is the energy eigenvalue always a real number?

Yes, the energy eigenvalue is always a real number. This is because the Schrödinger equation yields only real-valued solutions, and the energy eigenvalue is one of these solutions. However, in some cases, the energy eigenvalue may be negative, which can be interpreted as a bound state of the particle in the system.

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