Quantum mechanical integral equation problem

In summary: What are the expectation values of x and p for the given quantum mechanical particle?In summary, the conversation discusses the calculation of expectation values for a quantum mechanical particle described by the wave function Ψ(x). The participants suggest using gamma function integrals and a change of variable to simplify the integrals, resulting in an expectation value of x = x0/sqrt(a) and p = p0/sqrt(a). There is some confusion about the normalization of the wave function, but the general approach is agreed upon.
  • #1
bkmtkm
2
0

Homework Statement


The question is;
for a qunatum mechanical particle,
Ψ(x) = [1/(a1/21/4)].[e-(x-xo)2/2a].[eip0x/h]

in here, x0, p0 and h are constants, so,

Homework Equations


what are the <x>; expetation value, and <P>;expectation value of momentum ?

The Attempt at a Solution

,
[/B]
first we have to simplified to integral, than maybe can be resolvable
 
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  • #2
bkmtkm said:
first we have to simplified to integral, than maybe can be resolvable
So how can you express the expectation values in integral form?
 
  • #3
bkmtkm said:
The question is;
for a qunatum mechanical particle,
Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h
for the expectation value of x ,use gamma function integral
e.g ∫xnexp(-axm)dx=
(1/m)(Γ(n+1)/n)/a(n+1)/n
 
  • #4
I did this with change of variable, y=x-x0. gives integral of odd function = 0 plus standard integral (int exp(-y2/a) = sqrt(a)sqrt(pi)), result <x>=x0/sqrt(a). p is a little more tricky, but uses similar integrals, and the imaginary terms eventually cancel out giving <p>=p0/sqrt(a) - assuming that the 'constant h' is actually h bar.
 
  • #5
Erik 05 said:
I did this with change of variable, y=x-x0. gives integral of odd function = 0 plus standard integral (int exp(-y2/a) = sqrt(a)sqrt(pi)), result <x>=x0/sqrt(a). p is a little more tricky, but uses similar integrals, and the imaginary terms eventually cancel out giving <p>=p0/sqrt(a) - assuming that the 'constant h' is actually h bar.
There is a little problem here, because the original wave function is not correctly normalised. I think that it should be ##a^2## in the first exponential.
 
  • #6
I wondered that, similar examples I have seen have /2a2, so <x>=x0 and <p>=p0
 

1. What is a quantum mechanical integral equation problem?

A quantum mechanical integral equation problem is a mathematical problem that arises in quantum mechanics, a branch of physics that studies the behavior of matter and energy at a very small scale. It involves solving an equation that describes the relationship between the wave function of a quantum system and its potential energy.

2. Why is solving a quantum mechanical integral equation problem important?

Solving a quantum mechanical integral equation problem is important because it allows us to understand and predict the behavior of quantum systems, which are crucial in many fields such as chemistry, materials science, and electronics. It also helps us develop new technologies, such as quantum computers, that rely on the principles of quantum mechanics.

3. What are the challenges in solving a quantum mechanical integral equation problem?

One of the main challenges in solving a quantum mechanical integral equation problem is that it involves dealing with complex mathematical equations that are difficult to solve analytically. This often requires the use of advanced numerical techniques and computer simulations. Additionally, the behavior of quantum systems can be highly unpredictable, making it challenging to accurately model and solve the problem.

4. How do scientists approach solving a quantum mechanical integral equation problem?

Scientists approach solving a quantum mechanical integral equation problem by using a combination of theoretical and computational methods. This involves applying mathematical techniques, such as perturbation theory and variational methods, to simplify the problem and make it more tractable. Researchers also use advanced computer algorithms and simulations to solve the equations numerically and obtain accurate results.

5. What are some applications of solving quantum mechanical integral equation problems?

The solutions to quantum mechanical integral equation problems have many applications in various fields, such as developing new materials with specific properties, designing more efficient chemical reactions, and understanding the behavior of complex molecules. They also have practical applications in technologies like quantum cryptography, which relies on the principles of quantum mechanics to secure information.

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