Schrodinger equation Definition and 569 Threads

How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be...

Can anyone give me a really simple example on how to use the eqn above to solve it?

If there exists some normalized wavefunction ##\psi## that is not a solution to the Schrödinger equation (1D), what does this mean? You can still presumably use the square of the wave function to ascertain the probability it exists at some interval in space, but does it provide any other useful...

If I am trying to derive the energy eigenvalues and quantum numbers for the hydrogen atom (basic hydrogen-1), I obviously need to solve the hydrogen Schrodinger equation and account for some boundary conditions. However, no website ever gives me the boundary conditions. What would be the...

Homework Statement I am reading Mathematical Concepts of Quantum Mechanics (Stephen J. Gustafson, Israel Michael Sigal. Second edition). The book would like to find an evolution equation which would lead to the Hamilton-Jacobi equation $$\frac{\partial S}{\partial t}=-h(x, \nabla S) $$ in the...

Homework Statement Write down the general solution of the time-dependant schrodinger equation in terms of the solutions of the time-independant Schrodinger equation. Homework Equations TDSE TISE The Attempt at a Solution I'm really not sure how to interpret this question, I could write the...

The Hamiltonian operator in the equation i×h/2π×∂/∂t×ψ=H×ψ(where 'i' is the imaginary no.,'h/2π' is just expanded form of the reduced Planck constant,'∂/∂t' is the partial derivative with respect to time 't' and ψ is the wave function) is,as I recall,H=I+V(i don't know how to get those carets...

Homework Statement The position-space representation of the radial component of the momentum operator is given by ## p_r \rightarrow \frac{\hbar}{i}\left ( \frac{\partial }{\partial r} + \frac{1}{r}\right ) ## Show that for its expectation value to be real:## \left \langle \psi|p_r|\psi \right...

Is there any method for deduce Schrödinger equation from quantization of action??

Homework Statement Consider the full 1-electron hydrogen wave function. [/B] Prove that ψ =A(6r –r2/a0) exp[-^r3/a0] sinθ exp[+iφ], is a solution to the Schrodinger equation H|ψ> = E|ψ>, where H is the Hamiltonian operator. Hence show that it's energy E= -1.51 eV and its principle quantum...

why is psi = cos (k r - w t) + i sin ( k r - w t) = e^ [ i ( k r - w t)]? my question precisely is why not: 1. psi = sin (k r - w t) + i cos ( k r - w t) ? 2. psi = sin (k r - w t) + i sin ( k r - w t) ? 3. psi = cos (k r - w t) + i cos ( k r - w t) ? why not any of these three? is...

Hi, I am a senior year Physics undergraduate and my current understanding of quantum mechanics stands at the level of the Griffiths textbook. I am trying to understand what it means to renormalise the Schrodinger Equation. I know that it's not possible to understand the detailed mathematics of...

All explanations of Josephson effect I have read so far are based on Ginzburg–Landau theory. There seems no explanation based on Schrodinger equation. Why? While an explanation of Josephson frequency of 2eV/h seems not difficult to envisage, the major problem, I guess, should be with electron...

I've been having troubles resolving the Schödinger's time independent one-dimensional equation when you have a particle that goes from a zone with a constant potential to a zone with another constant potential, yet the potential is a continuos function of the form: $$ V(x)=\left\{...

I'm missing something obvious so please point out what I'm thinking wrong SE equation is: ih d/dt |> = H|> the taking adjoint turns i -> -i and (d/dt) -> -(d/dt) so adjoint of SE should be same as SE however it isn't. adjoint of SE is -ih d/dt |> = H|> do we not take adjoint of d/dt, if...

By solving the schrodinger equation, we get atmost two solutions for wavefunctions with definite wavenumber and definite wavelength. Thus, we know specifically the momentum of the particle. But this is contradicted by HUP. Please explain. I would appreciate an explanation in the context of a...

I am currently trying to learn a little about quantum mechanics, although not on very detailed level. There is one thing I wonder: What happens with the Schrödinger Equation in the classical limit, i.e. when either the mass of the particle tends to infinity or when Planck's constant tends to 0...

Homework Statement I am supposed to show that the free Schrödinger Equation is NOT kovariant under Galilei Transformation. Homework Equations We learned in Lectures that the Galilei Transformation can be written as: \vec{x'}=\hat{R}\vec{x}-\vec{a}-\vec{v}t (1) or equivalently...

While I am studying the wave propagation in fluids, the amplitude modulation seems to be governed by the Nonlinear Schrodinger (NLS) equation. In some of the journal papers the nonlinearity parameter, N seems to be of high value (N≈O(104)) and so on. I understand that weak nonlinearity...

This is a rather naive question concerning the dimension of the schrodinger equation. If the Schrodinger equation can be wrtiten in a three dimensional form using the laplacian operator can it be written in a 4d version. I understand that the schrodinger equation shows the development of the...

How to find wave function if potential function is unknown. I have only the scattering data: time and coordinates of scattering particle.

I got confused when in my book they went from one form of schrodinger equation to another. It doesn't make much sense to me algebraically, probably i have some lacks in complex numbers. Here are the equations: In the second one I think it's implied that above two equations give third and I...

I'm looking through my lecture notes, (studying relativistic corrections/perturbation theory using hydrogen), and I seem to have a mind block with one of the equations (the last one from the 3 in the middle). I know that the kinetic energy and coulomb potential has been subbed in for the...

I had a course of computational physics in university. When the professor wanted to non-dimensiolize the Schrodinger equation, among other things, he changed the wave function using the relation |\psi(x)|^2 dx=|\phi(y)|^2 dy where y is the non-dimensionalized postion (y=\frac x a) and so...

when Schrodinger equation is applied to SHO only positive value of potential energy changes it to Hermitian polynomial and hence solution is possible but potential energy is positive only when the particle is moving away from the the mean position.The sign of potential is negative when the...

Homework Statement Show that the wave function ##\Psi(x,t)=Asin(kx-ωt)## does not satisfy the time dependent Schrodinger Equation. Homework Equations ##-\frac{\hbar}{2m}\frac{\partial^2\psi(x,t)}{{\partial}x^2}+V(x,t)\psi(x,t)=i\hbar\frac{\partial\psi(x,t)}{{\partial}t}## The...

Homework Statement I'm trying to figure out how the general solution of the Schrodinger equation for a free particle when v=0 relates to anything I have learned in class...Homework Equations For Eψ=(hbar2/2m)d2ψ/dx2The Attempt at a Solution I really have no idea- what is confusing me is that ψ...

Homework Statement I need Part B of this question http://physics.wustl.edu/classes/FL2013/217/homework/ps03.pdf Recall that the free particle Schr¨odinger equation, i~ ∂ ∂tψ(x, t) = − ~ 2 2m ∂ 2 ∂x2 ψ(x, t) (1) has solutions of the “plane wave” form ψk(x, t) = exp[ikx − iω(k)t] , (2) where...

Homework Statement Trying to construct Shrodinger Equation given: * mass: m * Boundary Conditions: (potential) V(x)=-Vo exp(-x/L) for 0<x≤L V(x)=∞ for x≤0 Homework Equations The Attempt at a Solution (-h^2 / 2m ) (d^2 ψ / dx^2) + V(x)ψ = E * psi Not sure how to incorporate...

Hi, I am numerically solving the 2D effective-mass Schrodinger equation \nabla \cdot (\frac{-\hbar^2}{2} c \nabla \psi) + (U - \epsilon) \psi = 0 where c is the effective mass matrix \left( \begin{array}{cc} 1/m^*_x & 1/m^*_{xy} \\ 1/m^*_{yx} & 1/m^*_y \\ \end{array} \right) I know that...

In all the introductions to the Born-Oppenheimer approximation I've seen, they make the following claim: "If you write out the stationary Schrodinger equation for the simplest molecule -- H_2^+ -- even it cannot be solved analytically, so we are forced to make an approximation." But how do we...

Hi, I want to solve one dimensional Schrodinger equation for a scattering problem. The potential function is 1/ ( 1+exp(-x) ). So at -∞ it goes to 0 and at ∞ it's 1. The energy level is more than 1. I used Numerov's method and integrated it from +∞ (far enough) backwards with an initial value...

So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation. Since we are dealing with a free particle I can take the time independent equation, set V = 0...

Homework Statement Sketch the wave function ψ(x) corresponding to a particle with energy E in the potential well shown below. Show correctly relative values of amplitude and wavelength in different regions. Homework Equations none? The Attempt at a Solution I guess I was a bit...

The time dependent Schrödinger equation is: i\hbar\partial_{t}\Psi=\hat{H}\Psi Does it mean that the operator i\hbar\partial_{t} has the same eigenstates and eigenvalues as any Hamiltonian?

Hey guys, I just want to ask on how do you determine the form of wave function for Schrodinger equation of finite potential well and potential barrier. Why is it ψ(x) = Ae^ikx + Be^-ikx (x < -a) ψ(x) = Fe^ikx (x > a) ψ(x) = Ce^μx + De^-μx (-a < x < a) for k^2 =...

Homework Statement Consider the TISE for a particle of mass m moving along the x-axis and interacting an attractive delta function potential at origin: Part(a): What is the difference between a bound state particle and a free particle? Part(b): Show ##\psi _{(x)} = exp (-|k|x)## is a...

Homework Statement Here's something that's confusing me. Say we have a potential V(x) = Vo if x < 0, x > a and V(x) = 0 if 0 < x < a (yes I know the notation with greater than/equals etc isn't totally correct, but you know what I'm talking about.) In the middle section, ψ'' +...

This equation (see attachment) appears in one of Prof. Susskinds's lectures on Quantum Mechanics: in trying to differentiate the coefficients of the eigenvectors of a wave function with respect to time, an exponential e^(-iEt) is introduced for alpha. I can see that d/dt e^(-iEt) = -iE...

When normalising the S.E. in spherical coordinates you split it up into 3 integrals, with respect to r, theta and phi. My question is, once you have found the constants for each, when writing out the normalised PSI do you simply place them as a product in the solution? i..e PSI...

I'm going though the particle in a box lesson in my physics textbook right now. I understand all the math, but don't understand a lot of the physics behind it. Also this is an intro physics course, we're only covering the basics of quantum mechanics and not going into too much detail. How come...

To illustrate the abstract reduction to dimensionless quantities apply it to the harmonic oscillator V(x) = (m \omega^2 x^2) / 2 using x_0 = sqrt(h-bar/(m \omega)) and find a dimensionless Schrodinger equation. Translate the known solutions to the Schrodinger equation for the harmonic...

(If the equation below do not appear correctly, you can read all of the question in the attached file.) Solving the time dependent 1D Schrödinger equation, one can show that in all points (x,t), i\bar{h}\frac{\partial}{\partial t}\Psi(x,t)=-\frac{\bar{h}^2}{2m}\frac{\partial^2}{\partial...

Homework Statement Given \Psi(x, y, z)=(2/L)^{3/2}sin(\frac{n_x\pi x}{L})sin(\frac{n_y\pi y}{L})sin(\frac{n_z\pi z}{L}), calculate the first few energy levels and tell which are degenerate. The Attempt at a Solution I don't have much of an attempt to be honest.. What I've done so far...

I'm working my way through some QM problems for self-study and this one has stumped me. Given the Hamiltonian as H(t) = f(t)H^0 where f(t) is a real function and H^0 is Hermitian with a complete set of eigenstates H^0|E_n^0> = E_n^0|E_n^0>. Time evolution is given by the Schrodinger equation i...

Hello. The wave function or state vector (callled 'Ket') ψ in the time-dependent schrodinger equation i\hbar\frac{∂ψ}{∂t}=\widehat{H}ψ is the just energy eigenfunction or any wavefunction for the given system? For example, can ψ be momentum eigenfunction or angular momentum...

[b]1. How can I solve the Schrodinger equation for a potential V(x)= A sech^2(αx) ? How do I come to know that whether sech(αx) is a non-node bound state of the particular or not? [b]2. p^2/2m + V(x) = E [b]3. exp(kx)[A tanh(αx) + C]

Homework Statement The angular part of the Schrodinger equation for a positron in the field of an electric dipole moment {\bf d}=d{\bf \hat{k}} is, in spherical polar coordinates (r,\vartheta,\varphi), \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial...

i got a book on differential equations that says a shortcut to solving the general differential equation f'(x)+p(x)f(x)=g(x) is to take the antiderivative of g(x) dx times exp(-p(x) dx times x) to solve for f(x) where dx represents the functions antiderivative. (i kno its supposed to represent...

Hi Everybody, I am learning solid state physics using a German book called "Festkorperphysik" written by Gross and Marx. Now, in page 336 the Schrodinger equation in momentum space is introduced: \left( \frac{\hbar^2 k^2}{2m} - E \right) C_\vec{k} + \sum_\vec{G} V_\vec{G}...