Quick question on notation of the Hamiltonian

In summary, the conversation discusses the use of the upper and lower zeros in the context of degenerate perturbation theory and the Schrodinger equation. The upper zeros refer to the unperturbed Hamiltonian, while the lower zero represents the combination of the free Hamiltonian and potential term in the full Hamiltonian.
  • #1
rwooduk
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for a degnerate system it's in my notes that you can write:

[tex]H^{(0)}\Psi _{1}=E_{0}\Psi _{1}[/tex]
[tex]H^{(0)}\Psi _{2}=E_{0}\Psi _{2}[/tex]

and (not related) we write the general Schrodinger equation

[tex]H_{0}\Psi + V\Psi = E\Psi [/tex]

Please could someone tell me what both the upper and lower zeros on the H mean?

Thanks in advance
 
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  • #2
For the "upper zeroes", it seems to me its in the context of degenerate perturbation theory. Then that zero means its the unperturbed Hamiltonian.
For the "lower zero", the full Hamiltonian is written as the free Hamiltonian([itex] H_0 [/itex]) plus a potential(interaction) part ([itex] V [/itex]).
 
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  • #3
Shyan said:
For the "upper zeroes", it seems to me its in the context of degenerate perturbation theory. Then that zero means its the unperturbed Hamiltonian.
For the "lower zero", the full Hamiltonian is written as the free Hamiltonian([itex] H_0 [/itex]) plus a potential(interaction) part ([itex] V [/itex]).

Great thanks for clearing this up!
 

FAQ: Quick question on notation of the Hamiltonian

What is the Hamiltonian?

The Hamiltonian is a mathematical function used in classical mechanics and quantum mechanics to describe the total energy of a system. It is named after the physicist William Rowan Hamilton.

How is the Hamiltonian denoted?

The Hamiltonian is typically denoted by the symbol H. In quantum mechanics, it is often written as an operator, represented by the symbol Ĥ.

What is the significance of the Hamiltonian in physics?

The Hamiltonian plays a crucial role in understanding the dynamics of a physical system. It is used to calculate the equations of motion, and therefore, can provide information about the behavior and evolution of a system over time.

What is the difference between the classical and quantum Hamiltonian?

In classical mechanics, the Hamiltonian is a function of the position and momentum variables of a system. In quantum mechanics, it is represented by an operator that acts on the wave function of a system.

How is the Hamiltonian used in practical applications?

The Hamiltonian is used in a wide variety of applications, including calculating the energy levels of atoms and molecules, predicting the behavior of particles in accelerators, and studying the dynamics of complex systems in fields such as astrophysics and chemistry.

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