Eigenstates of the Hamiltonian

In summary: But anyways, the morale of the story: the Hamiltonian is not equal to the total energy. A system doesn't even have to carry a specific energy, but can be in a superposition.
  • #1
Master J
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When one says that a system is in an eigenstate of the Hamiltonian, what exactly does this mean?
I mean, if the Hamiltonian is the total energy of the system, then if it is in an eigenstate of the Hamiltonian, is this saying that its energy is a multiple of its total energy? Obviously this makes no sense. I hope you can see where I'm confused.
 
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  • #2
The Hamiltonian is not the total energy of the system. It is an operator. -- not a physical quantity. It is, however, referred to as the "energy operator".

If a system is in an eigenstate of the Hamiltonian, then the energy carries a specific energy, which is precisely the eigenvalue of the Hamiltonian.

If a system is in a superposition of eigenstates, then it simply doesn't carry a specific energy. Correspondingly, you can make some statements about the probability of measuring a specific outcome (the usual "coefficient squared").

But anyways, the morale of the story: the Hamiltonian is not equal to the total energy. A system doesn't even have to carry a specific energy, but can be in a superposition.
 
  • #3
Master J said:
When one says that a system is in an eigenstate of the Hamiltonian, what exactly does this mean?
I mean, if the Hamiltonian is the total energy of the system, then if it is in an eigenstate of the Hamiltonian, is this saying that its energy is a multiple of its total energy? Obviously this makes no sense. I hope you can see where I'm confused.

The thing that is an multiple of the state vector is actually the rate of change of the state vector. That's why the vector (state function) preserves its shape with the passage of time.
 
  • #4
If you have a list of eigenstates of the Hamiltonian, you essentially have a list of the possible energy levels you might find the particle in. Which one you'll find it in you won't know until you measure it.
 

1. What are eigenstates of the Hamiltonian?

Eigenstates of the Hamiltonian are the states in which the total energy of a quantum system is well-defined and constant. They are the solutions to the time-independent Schrödinger equation, and are represented by wave functions.

2. How are eigenstates of the Hamiltonian determined?

Eigenstates of the Hamiltonian are determined by solving the time-independent Schrödinger equation, which involves finding the eigenvalues and eigenvectors of the Hamiltonian operator. The eigenvalues represent the possible energies of the system, and the eigenvectors represent the corresponding states.

3. What is the significance of eigenstates of the Hamiltonian?

Eigenstates of the Hamiltonian are significant because they represent the stationary states of a quantum system, meaning the system will remain in that state indefinitely unless acted upon by an external force. They also play a crucial role in calculating the probabilities of different outcomes in quantum measurements.

4. Can a system have multiple eigenstates of the Hamiltonian?

Yes, a quantum system can have multiple eigenstates of the Hamiltonian, each with a different energy value. This allows for the existence of superposition states, where the system is in a combination of multiple eigenstates at the same time.

5. How do eigenstates of the Hamiltonian relate to energy levels?

Eigenstates of the Hamiltonian correspond to different energy levels of a quantum system. The lowest energy eigenstate is known as the ground state, while higher energy eigenstates are called excited states. The energy difference between these states is quantized, meaning the system can only transition between them in discrete steps.

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