A question on eigenstates and operators

In summary, we can say that the expression <x'|e^{i\hat{x}}|x> is equal to e^{ix'}\delta(x'-x), where \hat{x} is an operator. This can be proven using power series expansion. In general, if {\hat{Z}} is any operator and |Z> is an eigenfunction of this operator, we can also say that |Z> is an eigenfunction of e^{i\hat{Z}} with the corresponding eigenvalue of f(Z) for any function f(\hat{Z}).
  • #1
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Why can we say that:

[tex]<x'|e^{i\hat{x}}|x>=e^{ix'}\delta(x'-x) [/tex]

where where [tex]\hat{x}[/tex] is an operator?

I mean if

[tex]\hat{x}|x>=x|x> [/tex]

we may write [tex]<x'|\hat{x}|x>=x<x'|x>=x\delta(x'-x)[/tex]

but in the expression at the top, we have an exponential operator (something I've never come across before) - is |x> an eigenstate of this operator (it seems to be), and why is the eigenvalue of the operator exactly the same form as the operator?

Thanks.
 
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  • #2
You can prove it by using power series expansion

[tex]e^{i\hat{x}} = 1 + i\hat{x} - \hat{x}^2/2! + \ldots [/tex]

Eugene.
 
  • #3
meopemuk said:
You can prove it by using power series expansion

[tex]e^{i\hat{x}} = 1 + i\hat{x} - \hat{x}^2/2! + \ldots [/tex]

Eugene.

Aaahh thanks Eugene. So in general if [tex]{\hat{Z}}[/tex] is any operator and |Z> the eigenfunctions of this operator, we can also say that |Z> are also the eigenfunctions of [tex]e^{i\hat{Z}}[/tex].
 
  • #4
vertices said:
Aaahh thanks Eugene. So in general if [tex]{\hat{Z}}[/tex] is any operator and |Z> the eigenfunctions of this operator, we can also say that |Z> are also the eigenfunctions of [tex]e^{i\hat{Z}}[/tex].

That's right. |Z> is an eigenfunction of any function [tex]f(\hat{Z})[/tex] of the operator [tex]\hat{Z}[/tex]. The corresponding eigenvalue is [tex]f(Z)[/tex].

Eugene.
 

1. What is an eigenstate?

An eigenstate is a state in which a physical system is found with a definite value for a particular observable quantity. It is a state that is unchanged when acted upon by a particular operator.

2. What is an operator?

An operator is a mathematical function that operates on a physical system and produces a new state or value. In quantum mechanics, operators are used to describe the evolution of a system over time and to measure physical quantities.

3. How are eigenstates and operators related?

Eigenstates and operators are related through the eigenvalue equation, which states that the operator acting on an eigenstate produces the eigenvalue multiplied by the same eigenstate. In other words, the eigenstate is an eigenvector of the operator with the corresponding eigenvalue.

4. What is the significance of eigenstates and operators in quantum mechanics?

Eigenstates and operators are essential in quantum mechanics as they allow us to describe and understand the behavior of physical systems at the microscopic level. They help us calculate and predict the values of physical observables, such as energy and momentum, and understand the evolution of quantum systems over time.

5. How are eigenstates and operators used in real-life applications?

Eigenstates and operators are used in various real-life applications, such as in quantum computing, where they are used to manipulate and measure the state of qubits. They are also used in spectroscopy to analyze the energy levels of atoms and molecules, and in quantum chemistry to calculate molecular properties and reactions.

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