What is Schrodinger equation: Definition and 564 Discussions

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian.
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".

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  1. Z

    Are All Solutions of the 1D Schrodinger Equation Energy Eigenstates?

    Just like it says, are all solutions of the 1D time independent Schrodinger equation, by default, energy eigenstates? I'm having a hard time imagining how solutions, with these conditions, that aren't energy eigenstates could exist if they have to satisfy the relation E \psi(x)=\hat{H}\psi(x)
  2. RaulTheUCSCSlug

    I Normalizing Constant 3D Infinite Well

    For time independent Schrodinger's equation in 3-D Where Enx,ny,nz=(nx/Lx2+ny/Ly2+nz/Lz2)(π2ħ2/2m and Ψnx,ny,nz=Asin(nxπx/Lx)sin(nyπy/Ly)sin(nzπz/Lz) How do I normalize A to get (2/L)^3/2? I don't think I understand how to normalize constants.
  3. Z

    What makes localized energy eigenstates, localized?

    I'm reading about stationary states in QM and the following line, when discussing the time-independent, one-dimensional, non-relativist Schrodinger eqn, normalization or the lack thereof, and the Hamiltonian, this is mentioned: "In the spectrum of a Hamiltonian, localized energy eigenstates are...
  4. Schwarzschild90

    Planar wave solution to zero potential Schrödinger equation

    Homework Statement Homework Equations \begin{align} \begin{split} \psi(x, t) = e^{(ikx- i \omega t)} \\ V(x) = 0 \end{split} \end{align} The Attempt at a Solution For a free particle, the Schrödinger equation can be put in the form of ##\psi(x, t) = e^{(ikx- i \omega t)}##...
  5. Konte

    Variable separation - Schrödinger equation

    Hello everybody, My question is about variable separation applied in the solution of general time-independent Schrodinger equation, expressed with spherical coordinates as: \hat{H} \psi (r,\theta,\phi) = E \psi (r,\theta,\phi) Is it always possible (theoretically) to seek a solution such as...
  6. G

    Deriving hermite differential equation from schrødinger harm oscillator

    Homework Statement I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site: http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/ The Attempt at a...
  7. E

    Integrate Laplacian operator by parts

    This is the key step to transform from position space Schrodinger equation to its counterpart in momentum space. How is the first equation transformed into 3.21? To be more specific, how to integral Laplacian term by parts?
  8. W

    Schrodinger equation for potential step

    Homework Statement There is a stream of electrons with energy E, incident from x = -∞ on a potential step such that V(x) = ##V_{0}## for x<0 and 0 for x>0. E>##V_{0}##>0. Write the T.I.S.E for x<0 and x>0 and find the general solution for both. Homework EquationsThe Attempt at a Solution My...
  9. naima

    What is the law for probability having Schrodinger equation?

    I think that all is in the title. If the amplitude ##\phi## obeys a Schrodinger equation, what is the law for ##p = \phi^* \phi##?
  10. naima

    Is Fokker Planck equation equivalent to Schrodinger equation

    I found this question here but i do not understand the answer. One equation is from classical physics and the other from quantum physics. Is there another way to go from classical to quantics than the analytical extension?
  11. W

    How can the Schrodinger equation be used to solve a potential symmetry problem?

    Homework Statement (See attached photo) Homework EquationsThe Attempt at a Solution I Have no clue where to start for b), i need major help! For a I said that you can argue since the potential is symetric along the x-axis that this operator will go to 0. However I don't know where to start for...
  12. W

    Schrodinger equation in a well question?

    Homework Statement (See attached picture) Homework EquationsThe Attempt at a Solution I'm not even sure where to start :( I'm assuming that i have to find a wave function outside and inside the well and meet certain boundary conditions, but I am confused. Please help.
  13. M

    Time Independent Schrodinger Equation

    Homework Statement Verify that a plane wave ψ(x) = Ae-ikx is a solution to the time independent Schrodinger equation for a free particle in one dimension. Can it be normalised? Why?[/B]Homework EquationsThe Attempt at a Solution My lecturer's notes are all over the place, which is frustrating...
  14. RJLiberator

    How Does the Time-Dependent Schrödinger Equation Describe Qubit Evolution?

    Homework Statement Let |v(t)> ∈ℂ^2 by the time-evolving state of a qubit. If |v(0)> =\begin{pmatrix} 0 \\ 1 \end{pmatrix} , and the Hamiltonian of the system is H = \begin{pmatrix} 0 & -iλ \\ iλ & 0 \end{pmatrix} (where λ∈ℝ) what is |v(t)>?Homework Equations Time dependent schrodinger...
  15. A

    Schrodinger equation for step potential

    Homework Statement Consider the one-dimensional Schrodinger equation for the step potential, that is for U(x) = 0 for x<0, and for , . Consider a particle with mass m and energy E < U. Assume the particle is initially at x<0. a) Calculate the penetration depth Δx at which the probability...
  16. S

    Solution of Schrodinger equation in axially symmetric case

    The following extract is taken from Appendix A of the following paper: http://arxiv.org/abs/0810.0713.Any solution of the Schrodinger equation with rotational invariance around the ##z##-axis can be expanded as ##\psi_{k}=\Sigma_{l}A_{l}P_{l}(cos \theta)R_{kl}(r)##, where ##R_{kl}(r)## are the...
  17. A

    Schrodinger equation problem

    Homework Statement Consider the time-dependent Schrodinger equation for a free particle in two spatial dimensions Using the method of separation of variables, determine the wave function ψ(x,y,t) Homework EquationsThe Attempt at a Solution Not sure how to do the separation here since it is...
  18. entropy1

    Depiction of distribution of photons over time

    I have a question about photons and the Schrödinger equation. Photons behave like particles but also as waves. I understand that this can be described by the Schrödinger equation as a photon having a certain probability to be somewhere. If I understand this correctly, I take it that there are...
  19. D

    Griffiths Quantum Question. Unit-less Schrodinger Equation.

    In Grifftiths Intro to Quantum 2nd edition, page 51, he is re-expressing the Schrodinger equation for a harmonic oscillator in terms of a unit-less quantity \xi \equiv \sqrt{\frac{m\omega}{\hbar}}x So Griffiths takes the Schrodinger equation in equation [2.70] -\frac{\hbar^2}{2m}...
  20. Vinay080

    "Reading" Schrodinger's equation

    I am starting to learn Quantum mechanics. I can't wait for my completion of QM, as I am running behind all the concepts taught in the class; but I can't even go on studying chemistry, or I can't even analyse anything, without understanding the atoms in reality. I believe in (Russell's??)...
  21. U

    Clarification on Schrodinger's Equation

    Are my thoughts correct? **Wave function just means the wave function psi. I will specify when the wave function is squared. 1.) Schrodinger's Equation describes particles-their position, energy, spin (through the "numbers" l, n, and m). 2.) Simplified, SE says the total energy is the sum of...
  22. A

    Difference of Hydrogen Hamiltonian with relative mass particles

    Hi guys, I consider the qm-derivation of the electronic states of hydrogen. There are two different derivations (I consider only the coulomb-force): 1) the proton is very heavy, so one can neglect the movement 2) the proton moves a little bit, so one uses the relative mass ##\mu## The...
  23. A

    Finite difference Schrodinger equation

    I am simulating electrons inside a cylindrical well like the one shown on the first figure. My current work has been on solving the Schrodinger equation numerically for the above potential and then finding corrections to the solution such that it is consistent with Poissons equation. To do so...
  24. PhysicsKid0123

    Time dependent schrodinger equation and wave function

    1. Homework Statement p: momentum x: position t: time h_bar: Planck's constant Ψ: wave function Homework Equations The Attempt at a Solution I've posted a link to pictures. http://imgur.com/a/TKvUu I'm not vera good at using LaTex yet :( So I've shown that the wave equation satisfied the...
  25. A

    Numerical solution of Schrodinger equation script

    I am right now working on a script that solves the Schrodinger equation numerically for arbitrary potentials using the finite difference method. The idea is that I diagonalize the Hamiltonian with elements: H(i,i+1)=1/dx^2 * constants H(i,i-1)=1/dx^2 * constants H(i,i) = -2/dx^2 * constants and...
  26. A

    Schrodinger equation numerical solution

    I have written a program that solves the Schrödinger equation numerically using the finite difference method based on the attached article. The end goal is to make a program that solves the Schrödinger and Poisson equations self-consistently for the conduction band in different heterostructures...
  27. M

    Schrodinger equation - stationary states

    two questions: 1. besides using Ehrenfests theorem, is there another way of showing that the expectation value of momentum is zero in a stationary state ? (I don't see it when simply applying the definition on the stationary solution) 2. If we have a state that is a superposition of...
  28. Q

    Harmonic oscillator coherent state wavefunction

    Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw2x2/2 ) with eigenvalue of the lowering operator: b. I know you can do this is many ways, but I cannot figure out why this particular method does not work. It can be shown (and you can find...
  29. olgerm

    Analogues between QM- and CM N-body problem

    In CM general formulation of N-body problem is: x(N;D;T) = \iint \sum_{n=0}^{N_{max}} (\frac {(x(N;D;t)-x(n;D;t))*(m_N*m_n*G+q_N*q_n/(4*π*ε_0))}{(\sum_{d=0}^{D_{max}}((x(N;d;t)-x(n;d;t))^2))^{3/2}*m_N}) \, dt^2 Where x(N;D;T) is D´th coordinate of N´th body at time T. But to get equation of...
  30. D

    Scaling when solving Schrodinger equation numerically

    I guess this is just a maths problem about algebra. I'm learning to solve Schrodinger equation numerically, and right now I'm just dealing with the simplest examples like harmonic potential, square well, etc. The problem is that sometimes my program gives some strange results and I suspect it is...
  31. olgerm

    Schrödinger equation for 2 particles

    U(x,y,z,t)*ψ(x,y,z,t)-(ħ/(2*m))*(d2ψ(x,y,z,t)/dx2+d2ψ(x,y,z,t)/dy2+d2ψ(x,y,z,t)/dz2)=ħ*i*dψ(x,y,z,t)/dt qproton=-qe Schrödinger equation for electron in hydrogen atom (if we consider proton as point charge which is moving at a constant speed vproton→=(vp;x;vp;y;vp;z).) is...
  32. P

    Derivation of Time Dependent Schrodinger Equation

    Hi All, I have problem in understanding one step in the derivation of the time dependent Schrodinger Equation. Please see attached file page 2 (marked in red). Most grateful if someone can help! Peter Yu (This is from Quantum Mechanics The Theoretical Minimum Page 99-102)
  33. Dyatlov

    General solution for the time-dependent Schrödinger equation

    Hello! I have two uncertainties (hehe) about two concepts from a QM time-dependent Schrödinger equation video. The video is I cannot move on further if I don't fully grasp everything he explains in the video. My two issues are: 1) The general solution for the time-dependent Schrödinger...
  34. Avatrin

    Units of Wave function (Schrodinger equation)

    Hi I thought I knew the answer to this question until I encountered the following question: What is the unit of R(r)? We are of course talking about the radial part of the solution to Schrodinger's equation in spherical coordinates (i.e. \psi(r,\theta,\phi) = R(r)\Theta(\theta)\Phi(\phi)). I...
  35. S

    Boundary conditions of the radial Schrodinger equation

    Consider the radial differential equation ##\bigg( - \frac{d^2}{dr^2} + \frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_l (r) = \lambda\ \phi_l (r)##, which I've obtained by solving the Schrodinger equation in ##d## dimensions using the method of separation of...
  36. AwesomeTrains

    Calculating Force on a Particle using the Time Dependent Schrödinger Equation

    Hey! 1. Homework Statement I've been given the time dependent schrödinger equation in momentum space and have to calculate the force, as a function of the position, acting on a particle with mass m. \vec{F}(\vec{r})=-\nabla{V(\vec{r})}...
  37. wood

    Time independent Schrodinger equation and uncertainty in x

    Homework Statement Is the gaussian $$\sqrt{\frac{\pi}{2\alpha}}e^{-\alpha x^{2}}$$ an eigenfunction of ## \widehat{T} = \frac{\hat{p}^{2}}{2m}## ? If so, what is the corresponding eigenvalue? If not, find a P.E. operator ##\widehat{U} = U(\hat{x}) ## which gives rise to a Hamiltonian...
  38. AL-Hassan Naser

    Free electron or empty lattice schrodinger equation solution

    in the solution for free electron we start with X(x) = A exp (ikx) + B exp (-ikx) then using boundary conditions we eliminate B if the wave is traveling in the positive direction and vice versa my questions are: 1. what is the boundary condition used? 2. is it X(-inf) = 0? because this would...
  39. M

    Python, solving Schrodinger equation using Runge-Kutta

    Homework Statement I'm currently working on a project in which I have to solve the energy eigenvalues of the Schrodinger equation to compute the mass of certain Mesons. We've been taught very little programming (so apologies that my understanding is very basic), and are therefore given any...
  40. J

    Relativistic correction of Schrödinger equation

    Hello there, I've been given the relativistic correction of the Schrödinger equation for a free particle: $$ - \frac{\hbar^2}{2m} \frac{\partial ^2\Psi}{\partial x^2} - \frac{\hbar^4}{8m^3c^2} \frac{\partial ^4\Psi}{\partial x^4} + E_0 \Psi = i \hbar \frac{\partial \Psi}{\partial t} $$ How we...
  41. C

    Numerical solution to Schrödinger equation - eigenvalues

    Not sure whether to post this here or in QM: I trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfucnctions but I am confused about how exactly the eigenvalues come about. I've solved some differential equations in the...
  42. Rimmonin

    Is there a space-independent Schrödinger equation?

    I'm learning about the Schrödinger equation in one of my uni courses, and we've recently gone past how to solve the time-independent version. That got me wondering if there is a space-independent version of the Schrödinger equation and what it could possibly be used for. I know I'm probably...
  43. H

    What does the wavevector "k" mean in the Schrödinger eq. ?

    why the solution for energy levels of electron in 1D crystal lattice as solved in Kronig penny model has used wave vector k differently then the Schrödinger equation solved for a free particle. (only the conditions in the equation has changed not the maths...so the "USE" of wavevector 'k' must...
  44. quantumfunction

    How much time do you have to make the same measurement?

    As I undersand it, if you make a measurement on particle and get some observable property like spin, you can quickly make a second measurement and get the same outcome. How long do you have to make that second measurement before it starts evolving according to Schrodinger's equation and you...
  45. gfd43tg

    Time independent schrödinger equation query

    Hello, the TISE can be simplified $$H \psi = E \psi$$ Where ##H## is the Hamiltonian, and ##E## is the eigenvalue, but why don't the ##\psi## terms cancel, leaving ##H = E##? Also, what the heck does the eigenvalue ##E## have to do with the eigenvalue that I have previously encountered in...
  46. kini.Amith

    Separation of variables to solve Schrodinger equation

    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...
  47. T

    How can the Schrodinger equation be applied to a 3D rectangular box?

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

    Wavefunctions that don't satisfy Schrödinger equation

    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...
  49. S

    Boundary Conditions for Hydrogen Schrodinger Equation

    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...
  50. hunc

    Derive Schrodinger equation

    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...
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