Why can momentum be expressed as an operator

In summary, in quantum mechanics, momentum can be expressed as an operator, which is the eigenvector of the operator. This is determined by the 1D solution to the wave equation, where momentum is represented as -i\hbar \frac{\partial }{\partial x}. This is done through a variable substitution and is named "momentum" for convenience.
  • #1
bfusco
128
1

Homework Statement


Im having a hard time figuring out how in quantum mechanics things such as momentum can be expressed as an operator.

I know the simple algebra to get the relation. Starting with the 1D solution to wave equation[itex]\Psi=e^{i\omega x}[/itex] then differentiating that with respect to x and replacing the resulting k with the de Broglie relation you get [itex]\frac{\partial \Psi}{\partial x}=i\frac{p}{\hbar}\Psi [/itex].

From here I can sort of see that p may =[itex]-i\hbar \frac{\partial }{\partial x} [/itex] but I just don't get how a value such as momentum can be a operator.

Is it that to get the momentum from the wave function you need to perform this operator on the wavefunction, and since every term in the Schrodinger equation has the wavefunction in it, they simply represent momentum as the operator? Even if this is true, there has to be more to it than being a simple variable substitution right?
 
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  • #2
hi bfusco! :smile:
bfusco said:
… how in quantum mechanics things such as momentum can be expressed as an operator.

technically, momentum is the eigenvector of the operator, not the operator itself

however, the operator needs a name, so we call that "momentum" too! :rolleyes:

if this was biology … where they pay such detailed attention to nomenclature … we'd probably call it "momentumase" :wink:
 

1. What is the definition of momentum as an operator?

The momentum operator is a mathematical representation of the physical quantity of momentum in quantum mechanics. It is represented by the symbol "p" and is defined as the product of the mass of a particle and its velocity.

2. Why is momentum expressed as an operator in quantum mechanics?

In quantum mechanics, particles are described by wave functions that represent their probability of being in certain states. The momentum operator is used to describe the momentum of a particle in terms of these wave functions, allowing for the prediction of its behavior and interactions with other particles.

3. How is the momentum operator different from classical momentum?

Classical momentum is a physical quantity that is defined as the product of an object's mass and velocity. In quantum mechanics, momentum is expressed as an operator, meaning it acts on the wave function of a particle rather than representing a physical quantity that can be directly measured.

4. Can momentum be expressed as an operator for all types of particles?

Yes, momentum can be expressed as an operator for all types of particles, including both matter and non-matter particles such as photons. However, the specific form of the momentum operator may differ depending on the type of particle being described.

5. How is the momentum operator used in quantum mechanics calculations?

The momentum operator is used in quantum mechanics calculations to determine the momentum of a particle in a given state. It is also used in equations such as the Heisenberg uncertainty principle and the Schrödinger equation to describe the behavior and evolution of quantum systems.

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