Solve Eigenstate of i(d/dx): Find Finite Wave Function

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In summary, the conversation is about finding the eigenstate of the operator i(d/dx). The equation |psi|=exp [(a/i)x+C] was written and there was a question about omitting the || sign and whether there should be a negative sign before the exponent. A hint was given to think about functions with the same derivative as itself and to generalize it. It was also mentioned that exp(-i*ax+C) is finite for all real x.
  • #1
neelakash
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Homework Statement



I am to find the eigenstate of the operator i(d/dx)

Homework Equations



The Attempt at a Solution



After writing the eigenvalue equation with eigenvalue a, I got

|psi|=exp [(a/i)x+C] where C is the constant of integration...

Now, I do not understand what will be the result, if I omit the || sign from psi

The wave function has to be finite all the way-so,should there be any -ve sign before the exponent?
 
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  • #2
Can you show how you did it, |psi|=exp [(a/i)x+C] is very wrong in my opinion.

Hint: what function has the same derivate as itself?

Then generalize it.

And by the way, exp(-i*ax+C) is finite for all real x.
 

FAQ: Solve Eigenstate of i(d/dx): Find Finite Wave Function

What is an eigenstate in regards to solving i(d/dx)?

An eigenstate is a state in which a physical system can exist with a definite energy and other measurable properties, and it is a solution to the Schrödinger equation for a given system. In this case, the i(d/dx) operator represents the momentum operator, and the eigenstate solution will give us the finite wave function for a particle in that specific system.

How do you solve for the eigenstate of i(d/dx)?

To solve for the eigenstate of i(d/dx), we use the Schrödinger equation and set it equal to the energy eigenvalue multiplied by the eigenstate. Then, we solve for the finite wave function by taking the second derivative of the eigenstate equation and setting it equal to the energy eigenvalue multiplied by the eigenstate. This will give us a differential equation that we can solve to find the finite wave function.

Can you provide an example of solving for the eigenstate of i(d/dx)?

One example is solving for the eigenstate of i(d/dx) in the infinite square well potential. In this case, the energy eigenvalues are given by En = n^2*(h^2)/(8mL^2), where n is a positive integer and L is the width of the well. The corresponding eigenstates are psi_n(x) = sqrt(2/L)*sin(n*pi*x/L). By plugging these values into the Schrödinger equation and solving for the finite wave function, we can find the eigenstate of i(d/dx) for this system.

What is the significance of finding the eigenstate of i(d/dx)?

Finding the eigenstate of i(d/dx) allows us to determine the behavior and properties of a particle in a given system. It gives us information about the particle's energy and wave function, which can be used to make predictions about its behavior and interactions with other particles. This is crucial in many areas of physics, such as quantum mechanics and particle physics.

Are there any limitations to solving for the eigenstate of i(d/dx)?

Yes, there are limitations to solving for the eigenstate of i(d/dx). In some systems, the differential equations that result from the Schrödinger equation may not have analytical solutions, and numerical methods must be used to approximate the finite wave function. Additionally, the eigenstate solutions may not accurately describe the behavior of particles in highly complex systems, such as those involving strong interactions or relativistic effects.

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