Schrödinger Definition and 611 Threads

To start, we have: ψ=e^{i(kx+wt)} ψ_x=ikψ ψ_{xx}=-k^2ψ λ=\frac{h}{p} k=\frac{2π}{λ} E=\frac{p^2}{2m}+V Then, multiply both sides of energy equation by ψ to get: Eψ=\frac{p^2}{2m}ψ+Vψ And replace -k^2 in the wave equation with -\frac{p^2}{hbar^2}. Then plug into energy equation for p^2 to get...

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...

I guess this question can apply in all the generality of the 3D Schrodinger eqn. with a central force, the case I'm thinking of however is the the hydrogen atom. When solving the equation, we derive the quantization of the angular momentum, which has me thinking that before we begin quantizing...

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...

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 =...

I've come to the point in my homework discussing the above, and more specifically, energy levels, wave functions, excited states etc.. And while I can locate an appropriate formula and plug-n-chug, I'm finding that I have no clue what these equations and numbers actually mean. And(in my...

Hey, I've been searching around for papers reporting on the creation of relatively large cat states, the largest I have been able to find are by Wineland, and are on the scale of nano meters. Does anyone know of any articles where such states have been created (experimentally) and reported...

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'' +...

Hey there clever folk, So I am taking a class in Quantum Mechanics, and I've just completed an assignment about seperable solutions to the schrödinger equation. One of the questions was to show that seperable solutions give rise to time-independent probability densities. So I was having the...

Hi everyone, One approach to solve the Schrodinger equation is to use separation of variables: the solution is composed of a time dependant and space dependant component. When we go through the math, we get a time dependent LHS equal to a space dependant RHS, which means they must both be...

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...

Dear all, I meet a difficult question as follows: -\frac{1}{ρ^{2}}\frac{d}{dρ}(ρ^{2}\frac{dR_{l}}{dρ})+[\frac{l(l+1)}{ρ^{2}}+V(ρ)]R_{l}(ρ)=ER_{l}(ρ) (1) let x=ln(ρ) and y=ρ^{1/2}R_{l}(ρ) ,then y^{''}=γy...

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...

Homework Statement For the infinite square-well potential, find the probability that a particle in its fourth excited state is in each third of the one-dimensional box: (0 to L/3) (L/3 to 2L/3) and (2L/3 to L) Homework Equations ∫ψ^2= ProbabilityThe Attempt at a Solution So from ∫ψ^2 for...

(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...

given: at t=0 |PSI(0)> = 1/2 |PSI1> + (SQRT3)/2 |PSI2> --------------------------------------------------------------------- my attempt so far: we can write |PSI1> = 1/2 |UP> + 1/2 |DOWN> |PSI2> = (SQRT3)/2 |UP> + (SQRT3)/2 |DOWN> therefore |PSI(0)> = 1/2 |UP> + 1/2...

I'm using the Modern Physics by Tipler (6th edition) book. In sec 7.1 it talks about the first excited state being either E_(112 ) E_(121 ) E_(112). My question is what is the process of finding the n_(1),n_(2),n_(3) quantum numbers ? How i understand you pick random values and from their find...

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...

Homework Statement What is the expectation value of \hat{S}_{x} with respect to the state \chi = \begin{pmatrix} 1\\ 0 \end{pmatrix}? \hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}Homework Equations <\hat{S}_{x}> = ∫^{\infty}_{-\infty}(\chi^{T})^{*}\hat{S}_{x}\chi...

Homework Statement The evolution of a particular spin-half particle is given by the Hamiltonian \hat{H} = \omega\hat{S}_{z}, where \hat{S}_{z} is the spin projection operator. a) Show that \upsilon = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{-i\frac{\omega}{2}t}\\ e^{i\frac{\omega}{2}t}...

Hi, I've got a few questions about Schrödinger's formulation of QM, mostly about how to interpret the results: 1) How do I choose boundary conditions? I know that I should always normalize the wavefunction and make it continuous but some times you need to make the derivative continuous as...

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]

Hi, I'm reading www.phys.ethz.ch/~babis/Teaching/QFTI/qft1.pdf and trying to understand the canonical quantization of the Schrodinger field. In particular, the Lagrangian: \begin{equation} \mathcal{L} = \frac{i}{2}\psi^* \partial_0 \psi - \frac{i}{2}\psi \partial_0 \psi^* +...

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}...

Hopefully this is in the correct section. Struggling with this question though I don't think it should be particularly difficult: Show that if V(x) = V(-x) normalised solutions to the time-independent Schrodinger equation have definite parity - that is, u(x) = +-u(-x) (+- means plus or...

It is obvious to me how \hat {x} = x; \hspace{5 mm} \hat {p}_x = -i \hbar \frac {\partial} {\partial x} implies [ \hat {x} , \hat {p}_x ] = i \hbar and I can accept that these two formulations are mathematically equivalent, but I do not know how in general (or even in this specific...

Hi, I am confused about how we obtain a part of the Schrodinger equation for a particle of mass m that is constrained to move freely along a line between 0 and a. Equation: \frac{d^{2}ψ}{dx^{2}}+(\frac{8∏^{2}mE}{h^{2}})ψ(x)=0 Where does the value in the parenthesis come from and what...

We know that in Schrodinger equation, Ψ is called wave function, which is not observable, while Ψ·Ψ* is the probability, which is observable. can we rewirte the Schrodinger equation to a form without Ψ but only Ψ·Ψ*? because I think, in this way can I figure out all conservations in the...

How do I write a full Schrodinger equation, pre-approximation, for a mixture? Let's say 75% H2 and 25% He by number of particles. I already know the form and very basic applications of the Schrodinger equation and the Hamiltonian. What I want to know is, the specifics, such as how to specify...

I am reading David Griffiths' book on Quantum Mechanics, and he usually says that the general solution to the TDSE, given a potential V, can a DISCRETE linear combinations of the wavefunction solutions. However, in one section, he says that the linear discrete sum can be regarded as a continuous...

From RQM point of view, the Scrödinger cat appears 50 % times alive and 50 % times dead. From cat himself point of view he is 50 % times alive and 50 % times dead. But when it dies, he observe his reality like dead (non consciousness about external reality) or alive (conscious)-consciousness of...

In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't think page 262 is showing so I'll explain the gist of it: "In his initial paper" he considers...

Homework Statement I'm having a bit of trouble following my textbook, I was under the impression ψ(x) = e^i(kx) = Cos(kx) + iSin(kx) but in my textbook they write the general solution to this equation as ψ(x) = ASin(kx) + BCos(kx). How come they wrote the sin part as not imaginary? isn't this...

Homework Statement A particle of mass m is confined to move in a one-dimensional and Diract delta-function attractive potential V(x)=-\frac{\hbar^2}{m}\alpha\delta(x) where \alpha is positive. Integrate eh time-independent Schrodinger equation between -\epsilon and \epsilon. Let...

Should I start by learning about the equations for classical harmonic waves and how the de Brolier equations can be applied to them? What else should I learn? I'm a chemistry student and we did a class on quantum chemistry, but the mathematical side of it was way too complicated for me so I just...