Quantum Physics - hermitian and linear operators

In summary, the first part of the conversation discusses proving the Hermitian character of operators i(d/dx) and d^2/dx^2. The second part involves operators A and B, and the question of whether they are linear. The conversation also briefly mentions the momentum operator and the definition of linearity.
  • #1
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1. Prove that operators i(d/dx) and d^2/dx^2 are Hermitian.


2. Operators A and B are defined by:

A[itex]\psi[/itex](x)=[itex]\psi[/itex](x)+x

B[itex]\psi[/itex](x)=[itex]d\psi/dx[/itex]+2[itex]\psi/dx[/itex](x)

Check if they are linear.


The attempt at a solution


I noted the proof of the momentum operator '-ih/dx' being hermitian, should I just multiply all the terms involved in it by '-1/h'? I do not really know what should I do in the second exercise.
 
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  • #2
-ihbar d/dx is hermitean. You say you have the proof. Now dropping hbar which is a real (as opposed to an imaginary) constant, does it change the hermitean character or not ?

As for the second derivative operator, assuming wavefunctions dropping to 0 when going to infinity, can you show that it's hermitean by maneuvering the integrals ?

Consider the definition of linearity. It's not more complicated than that.
 

FAQ: Quantum Physics - hermitian and linear operators

What is the difference between hermitian and linear operators in quantum physics?

In quantum physics, operators are mathematical entities that represent physical observables such as position, momentum, and energy. Hermitian operators are those that have the property of being equal to their own conjugate transpose, while linear operators are those that preserve the linearity of a system. In other words, hermitian operators are self-adjoint, while linear operators are able to combine or scale multiple operators without changing the overall result.

How do hermitian and linear operators affect the measurement outcomes in quantum physics?

Hermitian operators are closely related to the concept of observables in quantum physics. When an observable is measured in a quantum system, the result will always be a real number. This is because hermitian operators have eigenvalues that are real numbers. On the other hand, linear operators can affect the measurement outcomes by changing the state of the system, but they will not change the actual values of the measurements obtained.

Can hermitian and linear operators be combined in quantum physics?

Yes, hermitian and linear operators can be combined in quantum physics. This is because both types of operators have mathematical properties that allow them to be combined without changing the underlying physical properties of the system. In fact, many physical observables in quantum mechanics are represented by a combination of hermitian and linear operators.

What is the significance of hermitian and linear operators in quantum physics?

Hermitian and linear operators play a crucial role in the mathematical formalism of quantum mechanics. They allow us to describe and understand the behavior of quantum systems, and they are essential for making predictions about measurement outcomes. Additionally, the properties of these operators provide insights into the fundamental concepts of quantum mechanics such as superposition and uncertainty.

Can hermitian and linear operators be applied to classical systems as well?

While hermitian and linear operators are primarily used in quantum mechanics, they can also be applied to classical systems. In classical mechanics, observables are represented by linear operators, while in quantum mechanics, observables are represented by hermitian operators. However, the mathematical properties of these operators remain the same in both classical and quantum systems, making them applicable in both contexts.

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