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. C

    Solving a Piecewise Schrodinger equation

    Homework Statement I was trying to solve the time-independent Schrodinger's equation for this well: http://i.imgur.com/C9QrvkX.png First I tried to look at cases where the energy of a particle is ##E < V_1##. Homework Equations Schrodinger's equation...
  2. The_Inventor

    A David Bohm's Paper on Hidden Variables Theory

    So I've been reading David Bohm's original paper on the alternative interpretation of quantum mechanics in terms of hidden variables, just out of interest. In the 4th section he presents a complex function ψ in terms of R and S, and then (using the time dependent schrodinger equation, TISE)...
  3. M

    Nonlinear Schrodinger Equation Dispersion Relation

    The Nonlinear Schrodinger Equation (NSE) is presented as: $$i\frac{∂A}{∂z} = \frac{1}{2}β_2\frac{∂^2A}{∂t^2}-\gamma|A^2|A$$ The steady state solution $$A(z)$$ Can be derived as an Ansatz given by: $$ A(z) = \rho(z)e^{i\phi(z)}$$ By substituting and solving the ODE, the steady state...
  4. SherLOCKed

    A Help with proof of eq. 2.64 of Intro. to Quantum Mechanics

    I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47. While the book has given a proof for eq. 2.64 but its not very ellaborate Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody...
  5. W

    I Time independent Schrodinger equation results (1D)

    okay so i need some help interpreting some of the results, so (-ħ2/2m)Ψ''=E-V0Ψ; So i set k2= 2m*(E-V0)/ħ2 and so : Ψ''=-k2Ψ so if V0=0 or is smaller than E, k2 is positive; *need for help starts here* Ψ=Aeikx+Be-ikx; another result for this would also be only eikx so is the second term only...
  6. D

    Question related to Schrodinger equation

    Homework Statement Homework EquationsThe Attempt at a Solution It is a short question so I don't suppose it is difficult. However, I don't really understand what it is asking for : 1.The TDSE itself is already a 2nd-order differential equation (if you substitute the terms back into H). 2...
  7. snate

    I Confused about complex numbers

    Can someone please explain what's going on at 47:40 Thanks in advance.
  8. woody stanford

    I Question about time-variant Schrodinger's eq'n

    The question I have is regarding the time-variant form of schrodinger's equation. Can I just put a complex number of form c=a+bi where the i is in it or can I just literally put sqrt(-1) where the i is: addendum: sorry forgot the t in the right-hand term, it should read (r,t) instead of (r)...
  9. yecko

    Schrodinger equation and Born's Rule

    Homework Statement [/B] Q18. Which of the following statements about Schrodinger equation is true? A) The exact solution of the equation never exists B) It is only applicable to the hydrogen-like atoms C) We can know the energy of the atomic orbital by solving the equation D) The square of the...
  10. mangojuice14

    Identical particles and separating the Schrodinger equation

    Homework Statement Two identical particles, each of mass m, move in one dimension in the potential $$V = \frac{1}{2}A(x_1^2+x_2^2)+ \frac{1}{2}B(x_1-x_2)^2$$ where A and B are positive constants and ##x_1## and ##x_2## denote the positions of the particles. a) Show that the Schrodinger equation...
  11. B

    I About general solutions to Schrodinger equation

    Hi, I am wondering why every general solution to Schrodinger equation can be built from separable solutions. In other words, I don't follow that why every solution to Schrodinger equation can be written as $$\Psi(x,t) = \sum c_n\Psi_n(x,t)=\sum c_n\psi_n(x)\phi_n(t)$$ I know that the right hand...
  12. J

    I Difference between statistical and dynamical properties

    Hi All, What are the main differences between statistical and dynamics properties in physics? Could you please explain the difference for problems in both classical and quantum mechanics. For instance, path integral molecular dynamics is supposed to give statistical properties of a quantum...
  13. P

    Solve time-dependent Schrodinger equation for V=V(x,t)

    Homework Statement For the potential ##V(x,t) = scos(\omega t)\delta (x) ## where s is the strength of the potential, find the equations obeyed by ##\phi_n(x)## And again for ##V(x,t) = \frac{\hbar^2}{2m} s \delta(x - acos(\omega t))## Homework Equations Time-Dependent Schro: ##...
  14. E

    A Is my solution of time-dependent Schrodinger equation right?

    The problem looks very simple. We have a time-dependent Hamiltonian: $$H(t) = B(t)H_0$$, where ##B(t)## is a numerical function, and matrix ##H_0## is time-indpendent. Let us consider: $$B(t) = \begin{cases} 1,&\text{for $0\leq t\leq t_0$}\\ A,&\text{for $t>t_0$.} \end{cases}$$ Also, let us...
  15. A

    I Numerical solution of Schrödinger equation

    Suppose I want to solve the Schrödinger equation numerically for some potential V(x). The easiest way to do so, is to discretize it on a grid of finite length, and apply a finite difference scheme to approximate the second order derivative. Doing so yields an eigenvalue equation on matrix form...
  16. S

    I Understanding Coherence in Quantum Mechanics

    Hi, Can anyone please explain the physical meaning of coherence(Quantum Mechanics). Thanks is advance
  17. M

    Solving Radial Schrodinger Equation

    Homework Statement This is a (long) multi-part question working through the various stages of solving the radial Schrodinger equation and as such it would be impractical to type it all out here but I will upload the pdf (https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU) of the...
  18. TheSodesa

    Showing that a wavefunction can be written as a product

    Homework Statement Let us look at a 3-dimensional potential box. Show, that the wave function in this situation can be written as the product of 3 single-argument functions. Homework Equations The 3D Schrödinger equation: \begin{equation} -\frac{\hbar^2}{2m} \left( \frac{\partial^2...
  19. K

    I Measuring Spin in the Stern Gerlach Experiment

    When we are measuring the spin of the electron in the experiment, we choose the spin property as its eigen state for the measurement. The eigen vectors corresponding to these states could be time dependent. Can we still break the problem into solving time independent Schrodinger Equation and...
  20. TheSodesa

    A relativistic electron in a potential box

    Homework Statement In a potential box (##L = 1.00pm##) an electron moves at a relativistic speed, meaning it's momentum can't be expressed as ##P = \sqrt{2mE}##. a) Using the uncertainty principle, show that the speed is indeed relativistic b) Derive an expression for the allowed energy states...
  21. Tspirit

    Wave function homework Problem 2.1 in Griffiths' book

    In the (b),I have some questions: (1) Does it mean ψ can be real or not real? (2) Why do the solutions of linear combination must have the same energy? As I know, these solutions are often different, as long as they are eigenvalues of time-independent Schrodinger equation. (3) In the sentence...
  22. T

    I Obtaining Spherical Harmonics to Normalized Angular Wave Functions

    The normalized angular wave functions are called spherical harmonics: $$Y^m_l(\theta,\phi)=\epsilon\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}*P^m_l(cos\theta)$$ How do I obtain this from this(http://www.physics.udel.edu/~msafrono/424-2011/Lecture 17.pdf) (Page 8)? The...
  23. A

    B Quantum field theory VS Quantum mechanics

    Hello I am little bit confused about one topic on theoretical Physics and that is If we want to describe our Quantum world (example atoms in metal) then should I use Quantum field theory or Quantum mechanics?
  24. A

    I Finite difference method for Schrödinger equation

    Suppose I want to solve the time-independent Schrödinger equation (ħ2/2m ∂2/∂x2 + V)ψ = Eψ using a numerical approach. I then discretize the equation on a lattice of N points such that x=(x1,x2,...,xN) etc. Finally I approximate the second order derivative with the well known central difference...
  25. Mayan Fung

    Physical interpretation of Schrodinger equation

    Schrodinger Equation is the very first step when we start learning QM. However, I never learned about the physical meaning of it. I have read a number of articles and discussion online. Regarding the ones I understand, there are generally two points of view. 1. Fundamental physical laws are not...
  26. Automata-Theory

    A Particle Moving Along a Ring with Variable Potential

    Homework Statement Alrighty, so here's my problem in a nutshell: Some particle of mass m is confined to move along a ring of radius R. Since it's on a ring, it has periodic boundary conditions--i.e.: For the boundary defined as ##-\pi R \leq x \leq \pi R##, ## x = -\pi R ## and ## x = \pi R...
  27. Titan97

    Normalized equation for particle in a ring

    Homework Statement Normalized equation for particle in a ring, where V=0 on a ring of radius 'a' and infinite everywhere else. Homework EquationsThe Attempt at a Solution Replcing x by rθ, $$-\frac{\hbar^2}{2I}\frac{\partial^2\psi}{\partial\theta^2}=E\psi$$ By guess, I found out that...
  28. A

    I How to discretize the Schrödinger equation with spin

    So I have previously learned how to discretize the Schrödinger equation on the form: (p^2/2m + V)ψ = Eψ , where the second order derivative is approximated as: (ψi+1+ψi-1-2ψi)/2Δx Such that the whole equation can be translated into a matrix eigenvalue-equation. The problem is that I am now...
  29. I

    I How to turn model of Schrödinger's Equation 2D?

    Hi, I am a student in the Netherlands, currently 17 years old and at the end of my 'middelbare school', meaning that next year I'll be a bachelor student at a university. I am doing an extended essay/research thing that is custom you do in your last year, with a friend of mine. We picked the...
  30. J

    I Just to check Schrodinger equation with 2 electrons

    Hi folks, I just want to check I understand correctly the Schrodinger equation for two electrons. https://en.wikipedia.org/wiki/Schrödinger_equation#Time-independent_equation With control F you can find "two electrons atoms or ions" section. Let's assume the wave function = x1 2+ x22 and...
  31. gimak

    Time independent schrodinger equation

    Homework Statement Lets say f(x) is a solution to TISE. If it is, why is its complex conjugate f*(x) a solution too? Homework Equations TISE = time independent Schrodinger equation The Attempt at a Solution ?
  32. P

    I How are different potentials implemented experimentally?

    Hi. I'm wondering how different potentials, such as the Dirac-Delta potential, linear potential, quandratic potenial, etc., are implemented experimentally. I only understand how the Schrodinger equation is solved if these are the potentials and I'd like to have a better understanding of quantum...
  33. DiracPool

    B Dirac Equation vs. Schrodinger Equation

    The Dirac equation is the more generalized form of the Schrodinger equation and accounts for relativistic effects of particle motion (say an electron) by using a second order derivative for the energy operator. If you have an electron that is moving slowly relative to the speed of light, then...
  34. F

    I Feynman Lectures: negative alpha for solving Schrödinger equation

    Why doesn't The Feynman Lectures consider the possibility of negative ##\alpha## when it says that ##e^{+2\alpha\rho}## is a rapidly increasing exponential (just below http://feynmanlectures.caltech.edu/III_19.html#mjx-eqn-EqIII1923) ?
  35. Q

    I Schrodinger equation in terms of complex conjugate

    I know there's a similar post, but i didn't understand it. Why the derivative respect to t in terms of the complex conjugate of ψ is: instead of being the original S.E in terms of ψ* or the equation in terms of ψ with the signs swapped
  36. F

    MATLAB Finite difference numerical integration or ode45?

    I'm trying to numerically solve the time dependent Schrödinger equation and I've been told that the best approach is to numerically integrate using a finite difference method, however I don't understand why I couldn't just use ode45 to solve it?! Is the finite difference (interpolation) method...
  37. T

    I Schrödinger equation and interaction Hamiltonian

    Given 1A.1 and 1A.2, I have been trying to apply the Schrödinger equation to reproduce 1A.3 and 1A.4 but have been struggling a bit. I was under the assumption that by applying ##\hat{W} \rvert {\psi} \rangle= i\hbar \frac {d}{dt} \rvert{\psi} \rangle## and then taking ##\langle{k'} \lvert...
  38. M

    Schrodinger equation for a weird potential

    Hello everyone, I have this weirdo potential for homework \begin{equation} U(x) = \frac{U_1}{ \left( 1+e^{x/a}\right)^2 } - \frac{U_2}{ \left( 1+e^{x/a}\right)} \end{equation} where U1,U2 and "a" are positive and I need to find the energies for the bound states and also the wave functions...
  39. N

    A Numerical solution to SE - variational method, many electrons

    Hi everyone, I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here. Problem I have can be explained on a very simple...
  40. J

    I Solving for SHM Diatomic Energy Levels

    So I'm trying to figure out how we got the allowed vibrational energy levels for a diatomic molecule by approximating it with simple harmonic motion. I do know how to use the uncertainty principle to get the zero-point energy: We know that the potential function is ##V(x) = \frac{1}{2}mx^2##...
  41. A

    Schrödinger Equation: Solving for Energy in a Semi-Infinite Square Well"

    Homework Statement I have an attachment Homework Equations Schrödinger equation The Attempt at a Solution The issues I am having is how to start this one. This is not a infinite square well but a semi-infinite square well. I know that energy= K^2= 2mE/h^2 Where h is planks constant 6.626 X...
  42. D

    Find the minimum kinetic energy of two electrons in a 1D box

    Homework Statement Problem: Consider a "crystal" consisting of two nuclei and two electrons arranged like this: q1 q2 q1 q2 with a distance d betweem each. (q1=e, q2=-e) a) Find the potential energy as a function of d. b) Assuming the electrons to be restricted to a one-dimensional...
  43. S

    A Integration of Schrodinger equation

    I'm trying to integrate the Schrodinger equation ##i\hbar \frac{d}{dt} |\psi(t)\rangle = H |\psi(t)\rangle## with the initial condition ##|\psi(t_{0})\rangle=|\psi_{0}\rangle## to show that ##|\psi(t)\rangle = \exp(\frac{t-t_{0}}{i\hbar}H)|\psi_{0}\rangle##. I know how to plug in the solution...
  44. R

    A The Nonlinear Schrödinger Equation

    According to my textbook the nonlinear Schrödinger equation: $$\frac{\partial A(z,T)}{\partial z} = -i \frac{\beta_2}{2} \frac{\partial^2A}{\partial T^2} + i \gamma |A|^2 A \ \ (1)$$ can be cast in the form $$\frac{\partial U(z,\tau)}{\partial z} = -i \frac{sign \beta_2}{2} \frac{1}{L_D}...
  45. J

    Angular momentum of hydrogen atom with Schrodinger Equation

    If we were to assume that the electron moves around the proton with radius a, the Schrodinger equation becomes: ##\frac{1}{a^2}\frac{d^2\psi}{d\phi^2} + \frac{2m}{\hbar^2}|E|\psi = 0## The question in my textbook asks me to solve the above equation to obtain values of energy and angular...
  46. S

    I Symmetry of Hamiltonian and eigenstates

    Suppose we have an electron in a hydrogen atom that satisfies the time-independent Schrodinger equation: $$-\frac{\hbar ^{2}}{2m}\nabla ^{2}\psi - \frac{e^{2}}{4\pi \epsilon_{0}r}\psi = E\psi$$ How can it be that the Hamiltonian is spherically-symmetric when the energy eigenstate isn't? I was...
  47. K

    Schrodinger Equation/verify solution

    Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a SolutionWe're...
  48. K

    IalChange of variables/verifying solution

    Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a SolutionWe're...
  49. amjad-sh

    I Schrodinger equation and stationary states

    Is the solution of the time-independent schrodinger equation always a stationary state? Can it be non-stationary?
  50. Danny Boy

    I Delta fuction potential general solution

    Hi, in the book 'Introduction to Quantum Mechanics' by Griffiths, on page 71 in the section 'The Delta-Function Potential' he states that the general solution to time independent Schrodinger Equation is $$\psi(x) = Ae^{-\kappa x} + B e^{\kappa x}$$ he then notes that the first term blows up as...
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