Eigenfunction Definition and 113 Threads

Homework Statement If y(x)=0 satisfies the ode and all the boundary conditions than does it count as the first eigenfunction? The Attempt at a Solution It wouldn't satisfy the orthogonality relation though? In that the integral of 0 and 0 is 0 even though the integral is over two...

I've been going round in circles with this problem for days: Find the eigenvalues and associated normalised eigenfunctions of the operator L: L_y = x^2 y'' + 2 xy' + \frac{y}{4} Boundary conditions y(1)=y(e)=0 So what I've done: substitute x = \exp(t) Then L_y = \frac{d^2y}{dt^2} +...

So, here's the question: \psi(x) = A*(\frac{x}{x_{0}})^n*e^(\frac{-x}{x_{0}}) Where A, n, and X0 are constants. Using Schrodinger's equation, find the potential U(x) and energy E such that the given wave function is an eigenfunction (we can assume that at x = infinity there is 0...

How does this method work? What are the mathematical ideas behind this method? Unlike separation of variables techniques, where things can be worked out from first principles, this method of solving ODE seems to find the right formulas and apply which I feel uncomfortable about.

The function sin(kx) is an eigenfunction of the KE operator? my work: KE=\frac{- \hbar^2}{2m}\frac{\partial^2}{\partial x^2} sin(kx) \frac{\hbar^2}{2m}\frac{sin(kx)}{k^2} i'm not sure how to show that a function is an eigenfuntion. what other work do I need to do?

which of the following functions is an eigenfunction of the momentum "operator" -i \hbar \frac{\partial}{\partial x}: f_1 =cos(kx- \omega t) f_2 =e^{a^2x} f_3 =e^{-(\omega t+kx)} for this question, I'm not sure what they are looking for... for f1 i \hbar k sin(k x -\omega t)...

The wavefunction of a particle moving inside a one dimensional box of length L is non-zero only for 0<x<L. The normalised wavefunction is given by: \psi (x) = \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L} Is this wavefunction an eigenfunction of the x-component of the momentum operator \vec p =...

Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction) Hi, I'm attempting to use Mathematica to calculate a mean-position eigenfunction of the Dirac equation. To do so I need to evaluate Fourier transforms (from p-space to r-space) of wavefunctions dependent on...

Hi! I encountered the problem that I need to decompose the Green function into a set of eigenfunction. Particularly, I have the free space Green function G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|} and I need to express it into series of...

I've yet to see a decent argument as to why the eigenfunctions of the position operator are delta functions. (Griffith's argues this, but oh so weakly). Could someone provide one or a couple dozen Kevin

we know that there's position eigenket,P eigenket and energy eigenket but is there something called the time operator,and the time eigenket or this is only a terminology that doesn't even exist,it's just my illusion! thanks gratefully

According to my book, uncertainty Q = 0 (where Q is an observable) is true when the state function is an eigenfunction. The energy eigenfunction for a particle in a 1-D box with infinitely high walls is sin(n*pi*x/a). This implies that the linear momentum, p, is known with zero uncertainty...

I am a little stuck understanding and answering the following questions. Can anyone help me with them? "A system has four eigenstates of an observable, with corresponding eigenvalues 3/2, 1/2, -1/2 and -3/2, and normalized eigenfunctions Psi_{3/2}, Psi_{1/2}, Psi_{-1/2} and Psi_{-3/2}...