Finding potential for a given wave function

In summary, Desh627 is trying to find the value for A that will satisfy normalization and determine the potential of the Schrödinger Equation using this value. They have not learned gaussian integration, which is hindering their progress. They have solved for A and need to find a technique to solve the integral of a Gaussian function. They also need to apply the time-dependent Schrödinger equation to determine the potential.
  • #1
Desh627
41
0
Here's the sitch:

I am given an equation, A*e-a(mx2/h-bar+it)

I need to find the value for A that will satisfy normalization, as well as find the Potential of the Schrödinger Equation using this value.

What do I do?

P.S. I have NOT learned gaussian integration, which is where I run into my first major problem, and this stops me from completing the problem.

Thanks guys,
Desh627
 
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  • #2
I should also mention what I have solved so far...

psi (x,t) = A*e-a(mx2/h-bar+it)

psi (x,0) = A*e-a(mx2/h-bar+i(0))
psi (x,0) = A*e-a(mx2/h-bar)

Integral from negative infinity to infinity dx (absolute value)A*e-a(mx2/h-bar)(/absolute value)2 = 1

A2 Integral from negative infinity to infinity dx e-a(mx2/h-bar)=1

A = sq. rt (1/Integral from negative infinity to infinity dx e-a(mx2/h-bar))

and this is where I run out of room.

I should also mention that I have taken high school physics and ap calc, but no linear algebra or anything beyond ap calc.
 
  • #3
Hi,
Let's see if we can't figure this one out.
To figure out how to normalize your wavefunction, you need to know what the integral of a Gaussian function is, where a Gaussian function is any function of the form:
[tex]f\left(x\right) = \exp\left(-\frac{x^2}{2\sigma^2}\right)[/tex]
I doubt you would know a good technique from AP calculus to solve this integral, so I would suggest reading this wikipedia article:
http://en.wikipedia.org/wiki/Gaussian_integral

For the second part, since your wavefunction already has time explicity in it, we can make a guess that it's a solution to the time dependent schrodinger equation:

[tex]\hat{H}\psi = i\hbar\frac{\partial}{\partial t}\psi[/tex]

where H is the operator such that:
[tex] \hat{H}\psi\left(x\right) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi\left(x\right) + V\left(x\right)\psi\left(x\right)[/tex]
And V(x) is the potential. Try applying the time-dependent schrodinger equation and see if you can convert the problem into a differential equation in x only (and not t). From there it might be possible to figure out what V(x) is.
 
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FAQ: Finding potential for a given wave function

What is "potential" in a wave function?

Potential in a wave function refers to the energy that is associated with a particle or system of particles. It is a measure of the force that the particles experience in a given environment, and it affects the behavior and properties of the wave function.

How do you determine the potential for a given wave function?

The potential for a given wave function can be determined by solving the Schrödinger equation, which is a mathematical equation that describes the behavior of quantum particles. The solution of this equation provides information about the potential energy of the system.

What factors influence the potential of a wave function?

The potential of a wave function can be influenced by various factors, such as the type and properties of the particles in the system, the external forces acting on the particles, and the boundary conditions of the system. These factors can affect the shape and magnitude of the potential energy.

Why is finding potential important in quantum mechanics?

Finding potential in a wave function is important in quantum mechanics because it allows us to understand and predict the behavior of quantum particles. The potential energy determines the stability and dynamics of the particles, and it is crucial in studying various phenomena such as particle interactions, energy levels, and wave behavior.

Can the potential of a wave function change over time?

Yes, the potential of a wave function can change over time. In quantum mechanics, the potential energy of a system can change due to external forces or interactions with other particles. This change in potential can cause the wave function to evolve and result in different outcomes or behaviors of the particles.

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