# Position operator is it communitative

• zimbabwe
If we consider the position operator \hat{x} and the Hamiltonian operator H, we can see that the two do not commute, as shown by the equation xH \neq Hx. This means that the order in which these operators are applied matters and they do not produce the same result if applied in a different order. This is known as noncommutativity and it is an important concept in quantum mechanics. In momentum space, the position operator is represented by \hat{x}=i\hbar\frac{\partial}{\partial p}, and in this space, it is also not commutative with the Hamiltonian operator. This shows that the position operator is not commutative in the Schroedinger Wave equation, and
zimbabwe
I was asked to show how the position operator is not communitative in the Shrodinger Wave equation. I thought it was as it is simply mulitplication

[x]=integral from negative to positive infinite over f*(x,t) x f(x,t) dx

Can anyone help shed some light on this. I may have misunderstood the question
$$\hat{x}=\int^{-\infty}_{+\infty}\Psi^{*}\left(x,t\right) x \Psi\left(x,t\right) dx$$

Last edited:
In momentum space $x$ is not commutative, because there

$$\hat{x}=i\hbar\frac{\partial}{\partial p}$$

So perhaps your professor is inquiring about the wave function in momentum space?

Show (that is my guess based on your information, which is rather few):

$$x H \neq H x$$

Integrals do not help for not commuting operators.

zimbabwe said:
I was asked to show how the position operator is not communitative in the Shrodinger Wave equation. I thought it was as it is simply mulitplication
A single operator can't be commutative or noncommutative on its own. You always have to talk about a pair of operators that commute or don't commute with each other. So the question we have is, the position operator does not commute with what? That's the extra information we need. (White assumed the missing operator was the Hamiltonian; jdwood83 assumed it was momentum...)

## What is the position operator and is it commutative?

The position operator is a mathematical operator used in quantum mechanics to determine the position of a particle in space. It is represented by the symbol x and is commutative, meaning that the order in which it is applied does not affect the result.

## How does the position operator work?

The position operator works by acting on a wave function, which describes the probability of finding a particle at a certain position. It multiplies the wave function by the position of the particle, giving the average position of the particle.

## Is the position operator commutative with other operators?

No, the position operator is only commutative with itself. It does not commute with other operators, such as the momentum operator, which is represented by the symbol p. This is known as the uncertainty principle in quantum mechanics.

## What is the significance of the position operator being commutative?

The commutativity of the position operator is important because it allows for the measurement of position and momentum to be independent of each other. This is essential in quantum mechanics, as it allows for the uncertainty principle to hold true.

## Are there any exceptions to the commutativity of the position operator?

Yes, there are exceptions to the commutativity of the position operator in certain cases. For example, in the case of a harmonic oscillator, the position operator does not commute with the Hamiltonian operator, which represents the total energy of the system. However, in most cases, the position operator is commutative.

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