What Is the Difference Between Hamiltonian and Hermitian Operators?

In summary, the conversation discusses the difference between the terms "hermitian" and "Hamiltonian" in the context of quantum mechanics. The main difference is that "hermitian" is a general mathematical property that applies to a wide range of operators, while "Hamiltonian" is a specific operator that encodes the dynamics of a quantum system. It is noted that a Hamiltonian must be hermitian, but not every hermitian operator is a Hamiltonian. The use of the term "hermitian" is also discussed, with some participants suggesting that "symmetric" and "self-adjoint" may be more accurate terms. It is also mentioned that the concept of the Hamiltonian extends from classical mechanics and is
  • #1
piareround
79
0
If anyone has time could they please answer this question.


I was looking and concept of the the http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" , I was wonder is their a difference between the two terms? If so how are Hermitian and the Hamiltonian different? Can anyone give an example?
 
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  • #2
"hermitian" is a general mathematical property which apples to a huge class of operators, whereas a "Hamiltonian" is a specific operator in quantum mechanics encoding the dynamics (time evolution, energy spectrum) of a qm system.

The difference should be clear.

A Hamiltonian must be hermitian, whereas not every hermitian operator is a Hamiltonian.

(the number 17 is positive number, but not every positive number is equal to 17 :-)
 
  • #3
'Hamiltonian' is also an adjective :-) (pertaining to Hamilton ?)

On a serious note, 'hermitean' is misleading. In the mathematics of quantum physics we should use <symmetric> and <self-adjoint>.
 
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  • #4
bigubau said:
... 'hermitean' is misleading. In ... quantum physics we should use <symmetric> and <self-adjoint>.
I fully agree!
 
  • #5
In finite spaces Hermitian works fine, but I agree that for infinite dimensional spaces self-adjoint is the better term.
 
  • #6
Also, the Hamiltonian is a concept that extends from Classical Mechanics. It is not unique to quantum mechanics
 

FAQ: What Is the Difference Between Hamiltonian and Hermitian Operators?

What is the difference between a Hamiltonian and a Hermitian?

A Hamiltonian is a mathematical operator that represents the total energy of a quantum mechanical system, while a Hermitian is a type of matrix or operator that has the property of being self-adjoint. In other words, a Hamiltonian is a specific type of Hermitian operator.

How are Hamiltonians and Hermitians used in quantum mechanics?

Hamiltonians and Hermitians are used to describe the dynamics and energy states of quantum systems. They play a fundamental role in the mathematical formalism of quantum mechanics and are used in various calculations and theoretical models.

Can a Hermitian operator be a Hamiltonian?

Yes, a Hermitian operator can be a Hamiltonian as long as it satisfies additional conditions, such as being bounded from below and having discrete eigenvalues. This means that not all Hermitian operators can be Hamiltonians, but all Hamiltonians must be Hermitian.

What is the physical significance of Hamiltonians and Hermitians?

Hamiltonians represent the total energy of a quantum system and can be used to predict the behavior and evolution of the system over time. Hermitian operators, on the other hand, have physical interpretations as observables in quantum mechanics, such as position, momentum, and energy.

Are there any other important differences between Hamiltonians and Hermitians?

Another important difference is that Hamiltonians are time-dependent, while Hermitians are not. This means that the eigenvalues and eigenfunctions of a Hamiltonian can change over time, while those of a Hermitian remain constant. Additionally, Hamiltonians are typically used in the Schrödinger equation, while Hermitians are used in other equations, such as the Heisenberg equation of motion.

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