False: The Time Independent Hamiltonian Operator "H

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In summary, the statement that the time independent Hamiltonian operator acting on an allowed state will give the same state back is false. This is because an "allowed state" refers to a physical state, which is not necessarily an eigenstate of the Hamiltonian. Therefore, the result of applying the operator will not always be the same state.
  • #1
mani5200
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Here is one of my test question. I did it wrong.

Specify whether the following statement is true or false and explain why? "The time independent Hamiltonian operator 'H' acting on an allowed state Psi will give the same state back, i.e "H Psi=E Psi, where E is the energy of the given state."

The answer is false. I don't know why but its false.

Can anyone explain me this?

Thank you.
 
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  • #2
What does allowed state mean? If it means one of the possible physical states of the system and therefore superposition states are "allowed" then the answer is obvious.
 
  • #3
An "allowed state" means a "physical state", and that is in general any (normalized) superposition of eigenstates of the Hamiltonian. So, a general state is not necessarily an eigenstate of the Hamiltonian.
 
  • #4
The answer should be "sometimes true, sometimes false" unless "allowed" means specifically that the state isn't an energy eigenstate.
 

1. What is the Time Independent Hamiltonian Operator "H"?

The Time Independent Hamiltonian Operator "H" is a mathematical operator used in quantum mechanics to describe the total energy of a system. It is a key component in the Schrödinger equation, which is used to find the wave function of a system at a specific time.

2. How is the Time Independent Hamiltonian Operator "H" different from the Time Dependent Hamiltonian Operator?

The Time Independent Hamiltonian Operator "H" is used to describe systems that do not change over time, while the Time Dependent Hamiltonian Operator is used for systems that do change over time. The Time Independent Hamiltonian Operator does not contain any time-dependent terms, while the Time Dependent Hamiltonian Operator does.

3. What is the significance of the Time Independent Hamiltonian Operator "H" in quantum mechanics?

The Time Independent Hamiltonian Operator "H" is an important tool in quantum mechanics because it allows us to calculate the energy levels and wave functions of a system. It also helps us understand how a system evolves over time and can be used to make predictions about the behavior of quantum systems.

4. How is the Time Independent Hamiltonian Operator "H" related to the concept of energy?

The Time Independent Hamiltonian Operator "H" represents the total energy of a system. It consists of kinetic and potential energy terms and is used to calculate the energy of a system at a specific time. It is a fundamental concept in quantum mechanics and is essential for understanding the behavior of quantum systems.

5. Can the Time Independent Hamiltonian Operator "H" be used for any type of system?

Yes, the Time Independent Hamiltonian Operator "H" can be used for any type of system, as long as it is in a time-independent state. This includes both simple systems, such as a single particle in a box, and more complex systems, such as molecules or atoms. However, for systems that do change over time, the Time Dependent Hamiltonian Operator must be used instead.

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