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".

View More On Wikipedia.org
Recent contents Top users
  1. R

    Time-independent Schrodinger equation in term of the TDSE

    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...
  2. Paradox101

    Schrödinger's time-dependent equation (general)

    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...
  3. R

    U(0)=0 for real expectation values of momentum

    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...
  4. G

    Method for deduce Schrödinger Equation

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

    Solution to the Schrodinger Equation

    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...
  6. AL-Hassan Naser

    Why Is the Wave Function Often Expressed as Psi = Cos(kr - wt) + i Sin(kr - wt)?

    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...
  7. S

    Renormalisation of the Schrodinger equation

    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...
  8. Z

    Is there any explanation of Josephson effect based on Schrodinger equation?

    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...
  9. A

    Solution to the Schrödinger equation for a non rigid step

    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\{...
  10. C

    Adjoint of Schrodinger Equation

    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...
  11. M

    Solutions of schrodinger equation contradicts HUP

    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...
  12. Erland

    Schrödinger Equation in the classical limit

    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...
  13. V

    Galilei Transformation Free Schrödinger Equation

    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...
  14. V

    On nonlinearity parameter in Nonlinear Schrodinger Equation (NLS)

    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...
  15. moriheru

    Exploring Dimensions: The 4D Schrodinger Equation Made Simple

    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...
  16. B

    Schrodinger equation with unknown potential function

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

    Different forms of Schrodinger equation

    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...
  18. D

    Any quick help with rearranging schrodinger equation in dirac notation

    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...
  19. ShayanJ

    Non-dimensionalization of Schrodinger equation

    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...
  20. D

    Application of Schrodinger equation to SHO

    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...
  21. M

    Time Dependent Schrodinger Equation

    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...
  22. 2

    General solution of the Schrodinger equation for a free particle?

    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...
  23. K

    Solution to Schrodinger Equation

    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...
  24. L

    Schrodinger Equation and 1D Box

    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...
  25. M

    Weak Form of the Effective Mass Schrodinger Equation

    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...
  26. A

    How do we *know* the Schrodinger equation for H2+ can't be solved?

    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...
  27. M

    Numerical solution of one dimensional Schrodinger equation

    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...
  28. E

    Solution to the 1D Free Schrodinger Equation

    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...
  29. A

    Wave function (schrodinger equation)

    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...
  30. M

    Left hand side of the Schrödinger equation

    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?
  31. M

    Form of Solution to Schrodinger Equation

    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 =...
  32. U

    Delta function potential; Schrodinger Equation

    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...
  33. M

    QM solutions to the Schrodinger Equation

    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'' +...
  34. M

    Generalized Schrödinger equation

    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...
  35. R

    Schrodinger Equation in Spherical co-ordinates. Constants.

    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...
  36. G

    Particle in a box Schrodinger equation

    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...
  37. A

    Dimensional Analysis - Schrödinger equation

    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...
  38. K

    Initial condition for Schrödinger equation

    (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...
  39. H

    Energy From Schrodinger Equation

    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...
  40. W

    The Schrodinger Equation: How to Solve for Time Evolution and Eigenstates

    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...
  41. G

    What is wavefunction in the time-dependent schrodinger equation?

    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...
  42. M

    Solving the Schrodinger Equation for V(x)=A sech^2(αx)

    [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]
  43. P

    Reducing angular Schrodinger equation to eigenvalue problem

    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...
  44. N

    Differential equations in the schrodinger equation.

    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...
  45. S

    Schrodinger equation in the reciprocal lattice.

    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}...
  46. C

    Schrodinger Equation for Constrained Particle

    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...
  47. P

    Can we rewrite Schrodinger equation using observable variable?

    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...
  48. B

    Schrodinger equation molecules

    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...
  49. P

    Solutions to Time-dependent Schrodinger Equation

    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...
  50. B

    Variational Derivation of Schrodinger Equation

    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...
Back
Top