Position operator in infinte vector space

In summary, you have successfully found an expression for <P|X|P> in terms of P(x) and P*(x) using the identity operator and the definition of the inner product.
  • #1
wakko101
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0

Homework Statement



Find an expression for <P|X|P> in terms of P(x) defined as <x|P> (and possibly P*(x) )

Homework Equations



X|P> = x|P>
Identity operator: integral of |x><x| dx

The Attempt at a Solution



Ok...<P|X|P> add the identity
= Integral [ <P|X|x> <x|P> dx ]
= Integral [<P|x|x> <x|P> dx ] Here I'm assuming x is a scalar and can be pulled out of the inner product
= Integral [ x <P|x> <x|P> dx ]
= Integral [ x <x|P>* <x|P> dx ]
= Integral [ x P*(x) P(x) dx ]

Does this look right?
 
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  • #2


Yes, your solution is correct. You have correctly used the identity operator to write <P|X|P> in terms of <x|P> and <x|P>*. This expression can also be written as <P|x> <x|P>, which is equivalent to x P*(x) P(x) as you have shown. Good job!
 

1. What is the position operator in an infinite vector space?

The position operator is a mathematical concept used in quantum mechanics to describe the position of a particle in an infinite vector space. It is represented by the symbol x and is used to calculate the position of a particle along a particular axis.

2. How is the position operator related to the wave function?

The position operator is related to the wave function through the position eigenvalue equation, which states that the position operator acting on the wave function gives the position of the particle in terms of the wave function. Mathematically, it is represented as xψ(x) = xψ(x).

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

The position operator and the momentum operator are both mathematical operators used in quantum mechanics, but they have different properties. The position operator measures the position of a particle, while the momentum operator measures the momentum of a particle. They also have different mathematical representations and do not commute with each other.

4. Can the position operator be used for particles in a finite vector space?

Yes, the position operator can be used for particles in a finite vector space as well. In fact, it is commonly used in classical mechanics to describe the position of particles in a finite space. However, in quantum mechanics, the position operator is more commonly used for particles in an infinite vector space.

5. How is the position operator related to the uncertainty principle?

The position operator is related to the uncertainty principle through the Heisenberg uncertainty principle, which states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This principle is represented mathematically by the commutator relation of the position and momentum operators.

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