# Momentum operator in quantum mechanics

• I
In summary, the momentum operator is represented differently in one spatial dimension and three spatial dimensions. In one dimension, it is represented as -iħd/dx and is not a vector operator. However, in three dimensions, it is represented as -iħ∇ and is a vector operator. In quantum mechanics, the momentum is represented by a vector operator in the position representation, and this follows the rules of vector transformations. This is not specific to quantum mechanics, as even in classical physics, momenta are represented as co-vectors.
The momentum operator for one spation dimension is -iħd/dx (which isn't a vector operator) but for 3 spatial dimensions is -iħ∇ which is a vector operator. So is it a vector or a scalar operator ?

In 3D the momentum is a (polar) vector and thus must be represented by a vector operator in quantum mechanics, and that's indeed the case as you correctly wrote: In the position representation,
$$\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla},$$
and this transforms indeed as a vector when operating on scalar fields (and the Schrödinger wave function is a scalar field under rotations!).

Your objection, that 3-d operators in 3-d can appear as scalar operators in 1-d, isn't really specific to QM.

Well, technically, as a covector :P

vanhees71 and weirdoguy
That's no quantum specific. Also in classical physics (canonical) momenta are co-vectors.

I know. I'm in a pedantic mood ;)

vanhees71 and weirdoguy

## 1. What is the momentum operator in quantum mechanics?

The momentum operator in quantum mechanics is a mathematical operator that represents the momentum of a quantum particle. It is denoted by the symbol "p" and is defined as the product of the mass of the particle and its velocity.

## 2. How is the momentum operator related to the wave function?

The momentum operator is related to the wave function through the momentum eigenvalue equation, which states that the momentum operator acting on the wave function yields a multiple of the wave function itself. This multiple is known as the momentum eigenvalue, and it represents the possible momentum values that the particle can have.

## 3. What is the significance of the momentum operator in quantum mechanics?

The momentum operator is significant in quantum mechanics because it is one of the fundamental operators used to describe the behavior of quantum particles. It plays a crucial role in determining the dynamics of a quantum system and is essential in calculating various physical quantities, such as the kinetic energy and the uncertainty in momentum.

## 4. How is the momentum operator represented in mathematical notation?

The momentum operator is represented in mathematical notation as a differential operator, given by p = -iħ∇, where ħ is the reduced Planck's constant and is the gradient operator. This notation indicates that the momentum operator operates on the wave function by taking the derivative of the wave function with respect to position.

## 5. How does the momentum operator behave under the principles of quantum mechanics?

The momentum operator behaves differently under the principles of quantum mechanics compared to classical mechanics. In quantum mechanics, the momentum of a particle is described by a probability distribution rather than a single definite value. This means that the momentum operator does not yield a precise value when acting on the wave function, but rather a range of possible values with associated probabilities.

• Quantum Physics
Replies
3
Views
2K
• Quantum Physics
Replies
12
Views
823
• Quantum Physics
Replies
3
Views
1K
• Quantum Physics
Replies
3
Views
544
• Quantum Physics
Replies
7
Views
741
• Quantum Physics
Replies
3
Views
1K
• Quantum Physics
Replies
14
Views
1K
• Quantum Physics
Replies
2
Views
1K
• Quantum Physics
Replies
6
Views
1K
• Quantum Physics
Replies
18
Views
901