Why do we need the position operator?

In summary, |Ψ|2 gives us the probability density of the wavefunction, Ψ, and integrating it gives us the probability of finding the particle at a desired location. The position operator, on the other hand, gives us an expectation value, which is different from a probability density. This is similar to the difference between a Gauss distribution and its corresponding average. The integrand Ψ*xΨ dx gives us the mean position.
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Phys12
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As I understand it, |Ψ|2 gives us the probability density of the wavefunction, Ψ. And when we integrate it, we get the probability of finding the particle at whichever location we desire, as set by the limits of the integration. But when we use the position operator, we have integrand Ψ*xΨ dx, which gives us the probability of finding the particle at a given place.

It seems to me as though, both of the above operations are doing the exact same thing. Then why do we have a position operator?
 
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  • #2
Phys12 said:
which gives us the probability of finding the particle at a given place
No. It gives an expectation value.

One is a probability density, the other is an expectation value. Different things ! Just like a Gauss distribution and the corresponding average are different things.
 
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The latter integrand will give you the mean position.
 
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BvU said:
No. It gives an expectation value.

One is a probability density, the other is an expectation value. Different things ! Just like a Gauss distribution and the corresponding average are different things.
Oh...I see, because it's the *expectation value*, got it. Thank you!
 

Related to Why do we need the position operator?

1. Why is the position operator necessary in quantum mechanics?

The position operator, also known as the x operator, is essential in quantum mechanics because it allows us to describe the position of a particle in three-dimensional space. Without this operator, we would not be able to accurately describe the behavior of particles on a microscopic level.

2. How does the position operator work?

The position operator works by operating on a wave function, which describes the quantum state of a particle. It returns a value that represents the position of the particle in space. This value can then be used to calculate other important properties, such as momentum and energy.

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

The position operator is significant because it is one of the fundamental operators in quantum mechanics. It allows us to make precise measurements of a particle's position, which is crucial in understanding the behavior of quantum systems. It also plays a key role in the Heisenberg uncertainty principle, which states that the more precisely we know a particle's position, the less precisely we can know its momentum.

4. Can the position operator be used to predict the exact position of a particle?

No, the position operator cannot be used to predict the exact position of a particle. According to quantum mechanics, the position of a particle is described by a probability distribution, rather than a definite value. The position operator allows us to calculate this probability distribution, but it cannot give us a precise position for a single particle.

5. How is the position operator related to other operators in quantum mechanics?

The position operator is closely related to other operators in quantum mechanics, such as the momentum operator and the Hamiltonian operator. These operators form a set of fundamental operators that are used to describe the behavior of quantum systems. The position operator also plays a key role in many other important concepts and equations, such as the Schrödinger equation and the wave-particle duality of matter.

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