Understanding Vector Spaces, Operators, and Eigenvalues in QM Homework

[ryanwilk]

Homework Statement

The vector |p> is given by the function x+2x² and the operator A = 1/x * d/dx, with x = [0,1].

a) Compute the norm of |p>

b) Compute A|p>. Does A|p> belong to the VS of all real valued, continuous functions on the interval x = [0,1]?

c) Find the eigenvalues and eigenvectors of A.

Homework Equations

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The Attempt at a Solution

a) For the norm, I have that it should be \sqrt{<p|p>} but I don't know how to find the scalar product of x+2x² with itself

b) A|p> is just 1/x + 4, which isn't continuous at x=0 so no?

c) I have no idea how to turn the operator into a matrix. Once I have the matrix, it should be easy but what is the matrix form of A = 1/x * d/dx? Or do I need to use the eigenvalue equation and say that A|p> = eigenvalue|p>?

Any help would be appreciated.
Thanks!

 

[dextercioby]

I sense somehow that the Hilbert space is L^2 \left(\left[0,1\right], dx\right), so computing the norm of that vector should be easy.

For the point c), you don't need any matrix, just the eigenvalue equation.

 

[ryanwilk]

I sense somehow that the Hilbert space is L^2 \left(\left[0,1\right], dx\right), so computing the norm of that vector should be easy.

For the point c), you don't need any matrix, just the eigenvalue equation.

Ah so for the norm, it's just \bigg(\int_0^1 (x+2x^2)^2 \mathrm{d}x\bigg)^\frac{1}{2}?

The eigenvalue equation will be \frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x} |\psi> = \lambda |\psi> but then how do you solve this, or do you assume that the eigenvector is |p>?

(Actually, |p> clearly isn't an eigenvector...)

 

[ryanwilk]

Oh wait, do you just solve it like a 1st order ODE to get |\psi> = Aexp\bigg(\frac{x^2}{2}\bigg) which means that the eigenvalue is 1?

 

[dextercioby]

Yes to the first post, kind of yes to to the second post when it comes to the wavefunction (actually \lambda should also be in the exponential), however, the spectrum of the operator (possible values of \lambda) is the entire complex plane.

 

[ryanwilk]

Yes to the first post, kind of yes to to the second post when it comes to the wavefunction (actually \lambda should also be in the exponential), however, the spectrum of the operator (possible values of \lambda) is the entire complex plane.

Oh there is a \lambda, I took it out because it looked a bit strange. So \lambda can be anything?

 

[dextercioby]

Yes.

 

[ryanwilk]

Yes.

Hmm interesting. Thanks a lot for your help!