Do All Physical States Satisfy the Hamiltonian Equation Hψ = Eψ?

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Thread starter Pyrus96
Start date Mar 28, 2016
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Hamiltonian Operator

In summary, the Hamiltonian operator is a mathematical operator used in quantum mechanics to represent the total energy of a system. It is related to the Schrödinger equation and is used to calculate the energy of a system at different points in time. The eigenvalues and eigenvectors of the Hamiltonian operator represent the allowed energy levels and corresponding wave functions of a quantum system. It can be used for systems with multiple particles and has various practical applications, such as modeling chemical reactions and developing quantum computing algorithms.

Mar 28, 2016

Pyrus96

I came across a previous exam question which stated: Do all physical states, ψ, abide to Hψ = Eψ. I thought about it for a while, but I'm not really sure.

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Mar 28, 2016

blue_leaf77

Science Advisor

Homework Helper

Pyrus96 said:

Do all physical states, ψ, abide to Hψ = Eψ

Not all, only eigenstates of the (time-independent) Hamiltonian satisfy that equation.

1. What is the Hamiltonian operator?

The Hamiltonian operator is a mathematical operator used in quantum mechanics to represent the total energy of a system. It is denoted by the symbol H and is defined as the sum of the kinetic and potential energy of all particles in the system.

2. How is the Hamiltonian operator related to the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. The Hamiltonian operator appears in the Schrödinger equation as an operator acting on the wave function of the system, allowing us to calculate the energy of the system at different points in time.

3. What are the eigenvalues and eigenvectors of the Hamiltonian operator?

The eigenvalues of the Hamiltonian operator represent the possible energy states of the system. The eigenvectors are the corresponding wave functions that describe these energy states. In other words, the eigenvalues and eigenvectors of the Hamiltonian operator tell us the allowed energy levels and the corresponding wave functions of a quantum system.

4. Can the Hamiltonian operator be used for systems with multiple particles?

Yes, the Hamiltonian operator can be used for systems with multiple particles. In such cases, the Hamiltonian operator will contain terms for the kinetic and potential energy of each individual particle, as well as terms for the interactions between the particles.

5. How is the Hamiltonian operator used in practical applications?

The Hamiltonian operator is used in various practical applications, such as modeling chemical reactions, studying the behavior of atoms and molecules, and developing quantum computing algorithms. It is also a key component in many theoretical models used to study complex systems in physics, chemistry, and other fields.