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. X

    One atom v.s. atoms? (solved by schrodinger equation.)

    The intense laser-atom physics becomes hot today. There is a famous interesting phenomenon: High-order harmonic generation (HHG). Lots of works are on the single atom response in the strong field approximation. Some of them obtain the spectrum by solving the Schrödinger equation. Then they say...
  2. K

    Schrodinger Equation Preserves Normalization

    Homework Statement Show that the time dependent Schrodinger equation preserves the normalization of the wavefunction 2. The attempt at a solution I don't need help showing this, I'm just not too sure what the question is asking. Do I just show that if a normalized wave-function \psi is a...
  3. L

    Schrödinger equation where V = |x|

    Hi there! I'm looking for the solutions of the stationary Schrödinger equation for a potential of the type V = |x| I know that the Airy functions are the solutions to the SE where V \sim x but for the above mentioned potential ... I can't find it -- neither in books nor on the net. Do...
  4. D

    Normalizing the Schrödinger Equation

    Hello all, How do you prove that, for the normalization of the Schrödinger equation, you can plug in the initial condition of psi(x,t-naught) that will satisfy normalization?
  5. Q

    Schrodinger Equation and Energy Quantization

    I'm a bit embarrased to ask this (thats why I'm asking here and not asking one of my professors), as a grad student in Physics I've had a good deal of quantum mechanics, but one thing I haven't fully understood yet is the mechanism in the Schrodinger Equation that forces eigenvalue quantization...
  6. A

    Numerical solution of Schrodinger equation

    Suppose for some specific problem (symmetric potential well) the Schroedinger equation is expected to give certain discrete bound states and corresponding eigenfunctions. Now I am trying to obtain the eigenfunctions by numerically solving the equation and plotting the solutions by randomly...
  7. A

    Relation between potential and bound states in Schrodinger equation.

    Suppose I have Schroedinger equation in the form: -u''(x)+V(x)u(x)=Eu(x) The potential is such that as |x| -> Infinity, V(x) reaches a constant positive value. In this case can we have bound state/plane wave solutions for u(x) with E > 0 ?
  8. Orion1

    Massless Schrödinger equation

    A massless particle situated in a 1D infinite square well with momentum only in the direction of quantum confinement (the x direction): E_t \psi (x) = \hbar \omega \psi (x) = i \hbar \frac{\partial}{\partial t} \psi (x) p(x) \psi(x) = \hbar k_x \psi (x) = -i \hbar \frac{\partial}{\partial x}...
  9. B

    What is the meaning of the lower case i in the Schrodinger equation?

    When examining the Schrodinger equation for a particle in a square well there is a lower case “i” that shows up in the equations that never gets defined. I have checked several sources and its usage is somewhat uniform and yet not defined. Can someone define the “i” ? One source I am...
  10. A

    Probability distribution of stationary Schrodinger equation

    Homework Statement Stationary Schrodinger equation for a particle moving in a potential well has two solutions psi1(x)=e^-ax^2 with energy E1 and psi2(x)xe^-ax^2 with energy E2 At t=o, the particle is in the state psi(x)=psi1(x)+psi2(x) Calculate the probability distribution as a...
  11. C

    Solution for schrodinger equation

    In the potential well example we are considering the potential in the well to be zero and infinite outside the boundary, does this mean that the electron can move freely such that there is no opposition or restoring energy acting on it. And also the probability of finding the electron is...
  12. B

    Momentum operator in Schrodinger equation

    Why momentum is replaced by momentum operator in Schrodinger equation ?
  13. C

    Schrodinger equation graph help

    I hope I'm not offending anyone here by posting a request for some help here. We are in need of a knowledgeable physicist who can interpret an anomoly in a Schrodinger equation graph. This is for an online alternate reality game ("ARG") called Find the Lost Ring. The graph can be found here...
  14. Z

    1-d Time Independent Schrodinger Equation Problem

    Homework Statement A particle of mass m and energy E, where E >V1 >V2 travels to the right in a potential defined as V(x) = V1 for - b < x < 0 V(x) = 0 for 0 < x < a V(x) = V2 for a < x < b (a) Write down the time-independent Schrodinger eq. and its general solution...
  15. C

    Solving the Time-Independent Schrödinger Equation for a Step Potential

    Considering a step potential of V(x) = o when x<o and V(x) = Vo when x>o so step occurring at origin of x axis. Write down in words the strategy for solving it. Answer: Solve the time-independent schrodinger equation for V=o when x<o and find the solution for the free particle wave function...
  16. C

    Time-dependant Schrödinger Equation Help.

    I have a project to work on that's due by mid March. I would need to write a computer program, to show how a wave packet reflect off a barrier? How much of the wave reflects off a wall of finite height and thickness, and how much tunnels through? I remember doing something similar in my...
  17. B

    What is the PDE for photons proposed by Bialynicki-Birula?

    Neither in Sam Treiman's http://books.google.de/books?id=e7fmufgvE-kC" I was able to find the Schrödinger equation for a photon, i.e. a particle without rest mass. The Schrödinger equation straight from Treiman's book (typos are mine, if any) -\frac{\hbar^2}{2m}\Delta\Psi + V\Psi =...
  18. T

    Schrödinger equation: eigen value or differential equation

    I have a problem on the basis of quantum mechanics and it's so simple that I'm almost too afraid to ask. Anyway: 1.) Schrödingers differential equation is used for time indepent and time depent problems. The solution is a wave function or the linear combination of the resulting wave...
  19. C

    Schrodinger Equation help

    Homework Statement The wave function of a particle satisfies the time-independent schrodinger equation. If the potential is symmetric and has the form V(x) = \inf |x|>1.0 V(x) = \frac{\hbar^2V_0}{2m} |x|<0.2 V(x) = 0 Elsewhere Using the shooting method, I need to find the...
  20. E

    Schrodinger equation and second derivatives

    Homework Statement Is this logic correct: begin logic; 1) The Schrodinger equation must hold for all x: 2) The Schrodinger equation contains a second derivative 3) A derivative is only defined on a continuous function. 1, 2, and 3 imply that the first derivative of any...
  21. D

    Physical Chemistry, P.W.Atkins, the Schrödinger equation

    Homework Statement Im doing an A-level project on the Schrödinger equation and am unsure on the mathematics used to obtain the following results: The Schrödinger for a particle in no potential field (=0) has the solution: psi(x)=e^ikx. i is defined below, I haven't really a clue as to...
  22. P

    Time-Dependent Schrodinger Equation

    Homework Statement Show that \Psi(x,t) = Ae^{i(kx-\omega t)} is a solution to the time-dependent Schrodinger equation for a free particle [ U(x) = U_0 = constant ] but that \Psi(x,t) = Acos(kx-\omega t) and \Psi(x,t) = Asin(kx-\omega t) are not. Homework Equations - \frac{h^2}{4\pi...
  23. A

    Schrödinger equation: macro level

    Is it possible, in theory, to describe a macroscopic object with the Schrödinger equation (its location for example)?
  24. R

    Another question on Schrodinger Equation

    I have read the following claim (where is not important): ...the Schrodinger equation provides no rational basis for the phenomenon of spin, the Pauli Exclusion Principle, or Hund's Rule... Is such a claim true ? If so, what does it matter ?
  25. P

    Schrodinger equation with Electric/Magnetic Potential

    Homework Statement Consider a particle of charge e and mass m in crossed E and B fields, given by E = (0,0, E), B = (0,B, 0), r = (x, y, z). Write the Schrodinger equation. Homework Equations Schrodinger's equation: \left[ -\frac{\hbar^{2}}{2m} \nabla^{2} + V(r,t) \right] \Psi(r,t)...
  26. H

    Slow variables in Nonlinear Schrodinger Equation

    Hi all. What do it mean by "slow variables" in NLS? I am reading a derivation of the NLS in the context of hydrodynamics, by R.S.John in his book "A modern introduction to the Mathematical Theory of Water Waves". In the book, slow variables are zeta = epsilon * (x-ct) and T = epsilon * t...
  27. E

    Dimensionless form of the time-independent Schrodinger equation

    For a free particle, show that the time-independent Schrodinger equation can be written in dimensionless form as d^2\psi(z)/dz^2 = -\psi(z) . I do not see how you would get rid of the m (with units mass) in front of the del in the SE (or the other constants for that matter)...
  28. E

    Solution to the Schrodinger Equation

    I am somewhat confused about how general the solution \psi_E(x) = Aexp(ikx)+Bexp(-ikx) is? Can someone complete one or both of these sentences: \psi_E(x) = Aexp(ikx)+Bexp(-ikx) is the solution to all equations of the form ... or \psi_E(x) = Aexp(ikx)+Bexp(-ikx) is a...
  29. R

    Where Can I Find Solutions for 3D Schrödinger Equation?

    Hello everyone! I am a new member and please sorry for some question that perhaps were discussed here before. But really I need your help. I am searching for methods of solving 3 dimensional Schrödinger equation. Till now in internet I coulnd't find any solution. All the papers and articles are...
  30. T

    Time independent schrodinger equation: delta potential

    I'm currently reading Griffith's book on time independent Schroedinger's equation about delta functions. However, I complete dislike how the book deals with the delta distribution. firstly, the book discusses how to solve: -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}-\alpha\delta\psi=E\psi for...
  31. B

    Can someone explain the Schrödinger equation?

    Sorry, I'm new, can someone provide a simplified (i.e. dumbed-down) explanation of the Schrodinger equation? i understand some of the basics of quantum mechanics, and i find this thread fascinating. at the base level, this seemed like a fairly straightforward question, but it seems there are...
  32. S

    Bound states for a Spherically Symmetric Schrodinger equation

    Homework Statement A particle of mass m moves in three dimensions in a potential energy field V(r) = -V0 r< R 0 if r> R where r is the distance from the origin. Its eigenfunctions psi(r) are governed by \frac{\hbar^2}{2m} \nabla^2 \psi + V(r) \psi = E \psi ALL in spherical coords...
  33. S

    Question about schrodinger equation

    Note that this is not homework... I am just curious The time independant Schrodinger equation can be written as \hat{H} \psi = E\psi IS there ever a time taht the above equation is not true?? what about the time dependant case?? We haven't gone over that in class so I am not quite...
  34. Q

    Schrodinger equation in the spherical coordinates

    If using spherical coordinates (r, theta, phi) , what is the meaning of the canonical momentum of theta, phi? What are their definitions and mathematical form? In solving the Hydrogen problem, one has not take into consideration P_theta and P_phi at all. Quantum River
  35. J

    Schrodinger equation Eigenfunction problem

    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...
  36. S

    Delta function potential and Schrodinger Equation

    I have a time dependent wavefunction for inside a delta function potential well: V(x) = -a delta(x). It reads Psi(x,t) = (sqrt(m*a)/hbar) * exp(-m*a*abs(x)/hbar^2) * exp(-iEt/hbar) I'm supposed to stick this back into the time dependent Schrodinger Equation and solve for E. Taking my...
  37. D

    Linearity of Schrodinger Equation

    Could someone please address this question? How do you algebraically demonstrate the superposition principle revealed by the Schrodinger equation (ie. If Psi1(x,t) and Psi2(x,t) are both solutions then Psi(x,t)= Psi1(x,t)+Psi2(x,t) is also a solution.)?
  38. S

    Schrodinger Equation HELP: Find Energy Emitted by Photon in Infinitely Deep Well

    HELP! I need to find the energy emitted by a photon if an electron is confined in an infinitely deep well. Can someone tell me what the hell to do please?! I'm desperate! Thanks!
  39. S

    Exploring the Time Dependence of Wave Functions with the Schrodinger Equation

    By noting that the time dependence of the wave function is governed by the Schrodinger equation show that \frac{d(\Psi^* x \Psi)}{dt} = \frac{i \hbar}{2m} \left[x\Psi^* \frac{d^2\Psi}{dx^2} - x \Psi \frac{d^2 \Psi^*}{dx^2} \right] not sure where to start on this one actually... do i start...
  40. E

    Schrodinger equation in gravitational field

    I read article http://www.oup.co.uk/pdf/0-19-850687-2.pdf There is described Schrodinger equation in gravitational field (1.10) and COW experiment. But once I found article, where it is shown that this Schrodinger equation is in direct contradiction with principle of equivalence. Can...
  41. W

    Struggling to Get the Same Result: Solving Schrödinger Equation

    In http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html" , he mentions this: I'm trying to get the same result, but I'm stuck. Has anyone done this before?
  42. D

    Complex Schrodinger Equation, references?

    Complex Schrodinger Equation, references?? hopefully i can explain what i am looking for well enough for somebody to understand. I am interested in finding any references for work that has been done on solving the schrodinger equation in C^n rather than in R^n, as in on the complex plane with...
  43. S

    Schrodinger equation in matrix form

    I have been asked to show that the Schroding equation is equivalent to: i(hbar)d/dt(cn(t))=sum over m (Hnm*cm(t)) where Hnm=integral over all space of (complex conjugate of psin)*Hamiltonian operating on psim psi=sum over n (cn(t)*psin) But i don't know how to even start this question.
  44. U

    Finding Probability of Particle in Box Length L w/Schrodinger Equation

    I think I copied the wrong notes or something because my notes do not follow. I am trying to find the probability of finding a particle in a box length L in the area \frac{L}{3}-\frac{\partial}{2} to \frac{L}{3}+\frac{\partial}{2} basically we have the following wave funtion...
  45. G

    You cannot derieve Schrödinger Equation .

    "You cannot derieve Schrödinger Equation". Bah. We're being told this over and over again. Then the game guy invents operators to extract momentum and energy from wavefunction, then puts them in Newtwon equation! He's saying exactly this: \frac{p^2}{2m} + V = E Should I look amazed when this...
  46. R

    A new nonlinear Schrodinger equation

    At this link: http://www.geocities.com/ptep_online/PP-04-07.PDF is a recent paper by Carlos Castro on a new nonlinear Schrodinger equation--for those that work in this area.
  47. K

    Schrödinger equation: P(r)>1 ?

    Schrödinger equation: P(r)>1 ? I have the solution of the Schrödinger equation for the ground state of the hydrogen electron. The solution ist: u100(r)=sqrt(1/(pi*a^3))*exp(-r/a) If I want to calculate some probabilty values I do this with: P(r)=4*pi*r^2*|u100|^2 If I set r=10^-13 I...
  48. M

    How Do You Find the Time Evolution of a Wave Function in Quantum Mechanics?

    I'm given the value of a normalized wave function at t=0 (see attachment) and I'm asked to find the wave function at some time t. I have no idea where to even begin, the book has zero examples of anything and I'm just stuck :confused:
  49. H

    Schrödinger equation and equivalence principle

    May be this is a silly question, but if one converts the nonrelativistic Schrödinger equation for a free particle to an uniformly accelerated frame, is the result the same as the Schrödinger equation for a particle within a gravitational potential? I was trying some simple calculations but did...
  50. Z

    Understanding the Terms in the 1-D Schrodinger Equation

    what is the physical significance of each of the terms in the 1-D time indipendant schrodinger equation?
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