QM: Hamiltonians and energies of a system

In summary, the conversation discusses the Hamiltonian of a system and whether the eigenenergies of the system follow the same form as the Hamiltonian. The conclusion is that yes, this can be done whenever the Hamiltonian is linear, and the wavefunction must satisfy certain conditions for it to work.
  • #1
Niles
1,866
0
1. The problem statement, all variables and given known data
Hi all.

Lets I have a system, whose Hamiltonian is given by [itex]H = p_1 + p_2[/itex], where pi is the momentum of i, where this i can be whatever, e.g. momentum in some particular direction. When the Hamiltonian is given on this form, do I know for certain that the eigenenergies of the system are on the form:

[tex]
E_{m,n} = f(m) + g(n),
[/tex]

where f and g are two functions that depend on respectively m and n? I.e. I am wondering if the eigenenergies of the system follow the same form as the Hamiltonian.

I hope you can help.Niles.
 
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  • #2
Niles said:
Lets I have a system, whose Hamiltonian is given by [itex]H = p_1 + p_2[/itex],

Shouldn't your Hamiltonian be quadratic in the momenta? Or are you doing relativistic QM in natural units?

When the Hamiltonian is given on this form, do I know for certain that the eigenenergies of the system are on the form:

[tex]
E_{m,n} = f(m) + g(n),
[/tex]

Yes, because Hamiltonians are linear operators. (Perhaps I should say that this can be done whenever the Hamiltonian is linear. I'm not sure if you can cook up a nonlinear one).
 
  • #3
Tom Mattson said:
Shouldn't your Hamiltonian be quadratic in the momenta?.

Yes, it should. Sorry for that.
Tom Mattson said:
Yes, because Hamiltonians are linear operators. (Perhaps I should say that this can be done whenever the Hamiltonian is linear ...

Hmm, can this be seen from the following?

[tex]
H \psi = E\psi \quad \Rightarrow \quad H_1 \psi + H_2\psi = E_1\psi + E_2\psi = E\psi.
[/tex]

Thanks for taking the time to reply.
 
  • #4
Yes. So here [itex]E=E_1+E_2[/itex].
 
  • #5
I've thought about this for the last couple of days. What I have found out is that I believe the above only works if the wavefunction satisfies [itex]
\psi(x_1,x_2) = \psi_1(x_1) \psi_2(x_2) [/itex], so it separates Schrödingers time-independent equation. Do you agree?
 

Related to QM: Hamiltonians and energies of a system

1. What is a Hamiltonian in quantum mechanics?

A Hamiltonian in quantum mechanics is a mathematical operator that represents the total energy of a system. It is a key concept in quantum mechanics and is used to describe the time evolution of a system.

2. How is the Hamiltonian operator represented mathematically?

The Hamiltonian operator is represented as H in mathematical notation. It is defined as the sum of the kinetic and potential energy operators of a system, H = T + V, where T is the kinetic energy operator and V is the potential energy operator.

3. What is the significance of the eigenvalues and eigenvectors of the Hamiltonian operator?

The eigenvalues and eigenvectors of the Hamiltonian operator represent the allowable energy states and corresponding wavefunctions of a system. The eigenvalues represent the possible energies that a system can have, while the eigenvectors represent the corresponding wavefunctions that describe the probability of finding a particle in a certain state.

4. How is the energy of a system calculated using the Hamiltonian operator?

The energy of a system is calculated by finding the eigenvalues of the Hamiltonian operator. These eigenvalues represent the different energy levels that a system can have, and the lowest eigenvalue corresponds to the ground state energy of the system.

5. Can the Hamiltonian operator be used to describe all physical systems?

The Hamiltonian operator is a fundamental concept in quantum mechanics and can be used to describe a wide range of physical systems, from simple particles to complex molecules. However, in certain cases, it may need to be modified to account for factors such as relativistic effects or interactions with external fields.

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