What is Schrodinger equation: Definition and 567 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. Tertius

    I Tunneling with Gaussian Wave Packet

    The goal is to have accurate 1D numerical results for tunneling probabilities through an arbitrary barrier without relying on analytic approximations such as WKB. If there is a more ideal approach to this, I am happy to change tactics. Time independent, for example, but I am not sure how to...
  2. Tertius

    Correct Setup for Finite Difference to Calculate Quantum Tunneling

    I thought I solved the problem in answering my own post a few days ago, but the tunneling probability vs. energy trend is clearly wrong. I've remade the post because I have totally changed my approach and need a better understanding of the boundary setup. Overall description: a plane wave...
  3. Tertius

    Numerical Solution to Schrodinger Equation w/ Coulomb Potential

    I am doing this to have my own solution for customization and understanding. I also want to manually check the WKB approximation accuracy at various energies against this static solution. I've split the problem into 3 regions and am solving it in 1D, but am having problems with how to define...
  4. J

    B New hypothesis of continuous wave function collapse

    In standard quantum mechanics, the wavefunction remains in a superposition of multiple possible states until it is "measured" or observed, at which point it collapses into one definite state. However, in this new model, there is no special role for measurement or observation. Instead, all...
  5. PhysicsRock

    I Derivation of normal Zeeman-Effect

    I was / am trying to derive the energy shift resulting from the normal Zeeman-Effect by coupling the Hamiltonian to the external field ##\vec{A}##, that carries the information about the field ##\vec{B}## via ##\vec{B} = \nabla \times \vec{A}##. Let ##q = -e## be the charge of the electron and...
  6. Ashphy

    Given the potential find the eigenfunction

    Hi, this was one of the oral exam questions my teacher asked so i tried to solve it. Consider y>0 the energy spectrum here is continuous and non degenerate while for y<0 the spectrum is discrete and non degenerate because E<0. for y>0 i thought of 2 cases case 1 there is no wave function for...
  7. Rayan

    Possible energy values given Hamiltonian

    So first I rewrote H as a matrix: $$ H = \begin{pmatrix} a & b \\ b & c \end{pmatrix} $$ And tried to find the eigenvalues/energies of H, so I solved $$ det (H - \lambda I ) = \begin{vmatrix} a-\lambda & b \\ b & c-\lambda \end{vmatrix} = (a-\lambda)(c-\lambda) - b^2 = ac - a\lambda -...
  8. richard_andy

    A Relation between the density matrix and the annihilation operator

    This question is related to equation (1),(3), and (4) in the [paper][1] [1]: https://arxiv.org/abs/2002.12252
  9. A

    A Schrodinger equation in quantum field theory

    What is the Schrodinger equation in QFT? is it the nonrelativistic approximation of a Klein-Gordon scalar field? or Is there more? I have read that the Schrodinger equation describes a QFT in 0 dimensions. I accept every answer
  10. hilbert2

    A The eigenvalue power method for quantum problems

    The classical "power method" for solving one special eigenvalue of an operator works, in a finite-dimensional vector space, as follows: suppose an operator ##\hat{A}## can be written as an ##n\times n## matrix, and its unknown eigenvectors are (in Dirac bra-ket notation) ##\left|\psi_1...
  11. L

    I Time-dependent to time-independent Schrödinger equation

    Why you can do separation of variables in time-dependent Schrödinger equation i \hbar \frac{\partial \psi(\vec{r},t)}{\partial t}=-\frac{\hbar^2}{2m}\Delta \psi(\vec{r},t)+V(\vec{r})\psi(\vec{r},t) with \psi(\vec{r},t)=\varphi(\vec{r})T(t) and when in general is that possible?
  12. Que7i

    Schrödinger Equation (why are U and -x^2 =0?)

    I've already found out how to do it, but, either I got lucky and my procedure is wrong or I just don't get it. Why are U and -x^2 equal 0?
  13. Samama Fahim

    I Schrodinger Equation from Ritz Variational Method

    (This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method) 1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
  14. D

    A practice problem with Schrodinger equation

    So my question is.. Is schrodinger equation for this problem like this?: How to use the condition that E=0? Thank you
  15. Mr_Allod

    Edge States in Integer Quantum Hall Effect (IQHE)

    Hello there, I am having trouble understanding what parts b-d of the question are asking. By solving the Schrodinger equation I got the following for the Landau Level energies: $$E_{n,k} = \hbar \omega_H(n+\frac 12)+\frac {\hbar^2k^2}{2m}\frac{\omega^2}{\omega_H^2}$$ Where ##\omega_H =...
  16. E

    Proof of Schrodinger equation solution persisting in time

    I've started reading Introduction to Quantum Mechanics by Griffiths and I encountered this proof that once normalized the solution of Schrodinger equation will always be normalized in future: And I am not 100% convinced to this proof. In 1.26 he states that ##\Psi^{*} \frac{\partial...
  17. gentzen

    A Dieter Zeh's MWI as Schrödinger equation + further assumptions

    Sorry for derailing that discussion even further. My reference to Dieter Zeh's German book was unlucky, not just because it is not a peer reviewed paper, but also because I did not remember the exact place with the remark. Since this has bothered me since a long time anyway, I now searched the...
  18. K

    I Schrodinger equation for N particles in a box

    [Pathria, statistical mechanics][1], pg2 ,when discussing ##N## particles in a volume ##V## "...there will be a large number of different ways in which the total energy E of the system can be distributed among the N particles constituting it. Each of these (different) ways specifies a...
  19. ThiagoSantos

    I Wave function using the time dependent Schrodinger equation

    Given a wavefunction ψ(x, 0) of a free particle at initial time t=0, I need to write the general expression of the function at time t. I used a Fourier transform of ψ(x, t) in terms of ψ(p, t), but, i don't understand how to use green's functions and the time dependent schrodinger equation to...
  20. Dario56

    I Kinetic Energy and Potential Energy of Electrons

    Time indepedendent Schrödinger equation for a system (atom or molecule) consisting of N electrons can be written as (with applying Born - Oppenheimer approximation): $$ [(\sum_{i=1}^N - \frac {h^2} {2m} \nabla _i ^2) + \sum_{i=1}^N V(r_i) + \sum_{i < j}^N U(r_i,r_j)] \Psi = E \Psi $$ Terms in...
  21. Hamiltonian

    I Derivation of Schrodinger equation (chicken and egg problem?)

    The classical wave equation in 1-D reads: $$\frac{\partial^2 u}{\partial x^2}(x,t) = \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}(x,t)$$ The D'alembert solution to the wave equation is: $$u(x,t) = f(x+vt) + g(x-vt)$$ so a allowed wave function solution to the 1-Dimensional classical wave...
  22. kbansal

    How to explain the Quantum Mechanics/Math of the stages of MRI imaging

    "B0 is a static magnetic field (produced by a superconducting magnet) that initially causes the protons in the body to align with the field and precess at the larmor frequency along the z axis . From a mathematical perspective this precession around the B0 axis occurs due to the time evolution...
  23. D

    I Physical interpretation of phase in solutions to Schrodinger's Eqn?

    Hello all, So I've been working through the solutions to some simple introductory problems for the Schrodinger Equation like the infinite square well, and I'm trying to make sense of how to think about the phase component. For simplicity's sake, let's start off by assuming we've measured an...
  24. Wannabe Physicist

    Exploring the Time-Energy Uncertainty Relation to Solve a Schrodinger Equation

    I am guessing time-energy uncertainty relation is the way to solve this. I solved the Schrodinger equation for both the regions and used to continuity at ##x=-a, 0,a## and got ##\psi(-a<x<0) = A\sin(\kappa(x+a))## and ##\psi(0<x<a) = -A\sin(\kappa(x-a))## where ##\kappa^2 = 2mE/\hbar^2##...
  25. thegroundhog

    I Proving the Schrodinger Equation

    How did scientists prove the accuracy of Schrodinger's equation to describe the behaviour of subatomic particles, especially in the 1920s? How do you monitor an electron's momentum and position when they are so small? Also, if the Schrodinger equation just describes probabilities, is the...
  26. L

    I Schrodinger Equation as Flow Equation

    I was playing with the Schrodinger equation and realized that it can be interpreted as a flow equation. If we set $$ \psi = A e^{i \theta} $$ We can put the Schrodinger in the form ∂ψ∂t=(−∇ψ)⋅v+iEψ If v=ℏθm and E=ℏ2m(−∇2AA+∇2θ)+ρV I find this intuitive personally as it shows that the...
  27. C

    Solving 1-D Schrodinger Equation in Python (Scipy) Numerically

    I've tried to make an animation using python to demonstrate the 1-D simple harmonic oscillator and step potential examples. Hope that it can be useful for some of you. Have fun~ :) https://blog.gwlab.page/solving-1-d-schrodinger-equation-in-python-dcb3518ce454 By the way, If you are...
  28. O

    I General solution of the hydrogen atom Schrödinger equation

    Hello everyone! I have two questions which had bothered me for quite some time. I am sorry if they are rather trivial. The first is about the general solution of the hydrogen atom schrödinger-equation: We learned in our quantum mechanics class that the general solution of every quantum system...
  29. Huzaifa

    Explain the Schrodinger equation

    Please explain in simple words, the meaning of the Schrodinger wave equation in the quantum mechanics model of atom. $$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}+\frac{8 \pi^{2} m}{h^{2}}(E-U) \psi=0$$
  30. P

    I Why the linear combination of eigenfunctions is not a solution of the TISE

    The linear combination of the eigenfunctions gives solution to the Schrodinger equation. For a system with time independent Hamiltonian the Schrodinger Equation reduces to the Time independent Schrodinger equation(TISE), so this linear combination should be a solution of the TISE. It is not...
  31. F

    Current through Ballistic 2DEG Channel

    So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is: 1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system: -Here I struggle a bit with actually...
  32. Tuca

    I Solving Schrödinger's equation for a hydrogen atom with Euler's method

    Hi, first-time poster here I'm a student at HS-level in DK, who has decided to write my annual large scale assignment on Schrödinger's equation. My teacher has only given us a brief introduction to the equation and has tasked us to solve it numerically with Euler's method for the hydrogen atom...
  33. S

    I Matrix Notation for potential in Schrodinger Equation

    I'm working on the time-dependent Schrodinger equation, and come across something I don't understand regarding notation, which is not specific to TDSE but the Schrodinger formalism in general. Let's say we have a non-trivial potential. There is a stage in the development of the TDSE where we...
  34. fresh_42

    AI meets Chemistry in solving electronic Schrödinger equation

    Abstract: The electronic Schrödinger equation can only be solved analytically for the hydrogen atom, and the numerically exact full configuration-interaction method is exponentially expensive in the number of electrons. Quantum Monte Carlo methods are a possible way out: they scale well for...
  35. E

    Schrodinger Equation Research Question

    Hey guys, I have had my eye on quantum mechanics for a while now and finally decided that I have a large enough understanding of the concept/math/theories behind it to write a research paper on it, specifically Schrodinger Equation. But I am having a hard time finding a good research question...
  36. LCSphysicist

    Schrodinger equation in three dimensions. Atom with one electron.

    When solving the Schrodinger equation by separation of variables to atom with one electron and in the spherical coordinates, we get $$\Psi = \Theta(\theta)\phi(\varphi)R(r)$$ Specifically, $$\phi = e^{im\rho }$$ The question is, why we adopt this particular solution, in general, we have this...
  37. A

    Exponential Wavefunction for Infinite Potential Well Problem

    Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0. I set up my normalization integral as follows: A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1 After simplifying, and accounting for the fact that...
  38. redtree

    I Relativistic quantum mechanics

    Given that the Minkowski metric implies the Lorentz transformations and special relativity, why do the equations of relativistic quantum mechanics, i.e., the Dirac and Klein-Gordon equations, require a mass term to unite quantum mechanics and special relativity? Shouldn't their formulation in...
  39. R

    I How do you normalize this wave function?

    I have a basic question in elementary quantum mechanics: Consider the Hamiltonian $$H = -\frac{\hbar^2}{2m}\partial^2_x - V_0 \delta(x),$$ where ##\delta(x)## is the Dirac function. The eigen wave functions can have an odd or even parity under inversion. Amongst the even-parity wave functions...
  40. Ssnow

    I Schrodinger equation on the complex disk

    Hi to all member of the Physics Forums. I have this question: it is possible consider the analogue of the Schrodinger equation on the plane with configuration space ##(x,p)\in\mathbb{R}^4## on the complex disk ##\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}##? Ssnow
  41. K

    TISE solution for a hydrogen atom

    I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck: $$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...
  42. ShadownightPrograms

    I Many-Worlds Theory: Simulation Program in C#

    Here is the Code File in an txt. I can on request provied the whole Program, which includes the PSE, AtomFunctions and many useful but not all implemented Funtions to solve the Many Worlds Problem in C#. Please feel free to ask questions via here or email [e-mail address deleted by Mentors] I...
  43. JorgeM

    Gaussian wavepacket as a solution of the Schrödinger equation

    The Schrödinger equation I need to prove is this one And the Gaussian wavepacket is found here Thanks for your advice. JorgeM <Moderator's note: upload images to PhysicsForums. Do not use external image servers.>
  44. patric44

    A problem in solving Schrodinger equation for hydrogen

    hi guys i am having a little problem concerning the theta part of TISE : its clearly that its very similer to the associated Legendre function : how iam going to change 1/sinθ ... to (1-x^2) in which x = cosθ i tried many identities but i am stuck here . any help on that ?
  45. Boltzman Oscillation

    Solving Schrodinger's Equation for a Particle in an Infinite Box

    Firstly, since there is no condition for the z axis in the definition of the potential can I assume that V(x,y,z) = .5mw^2z^2 when 0<x<a, 0<y<a AND -inf<z<inf? If so then drawing the potential I can see that the particle is trapped within a box with infinite height (if z is the...
  46. Diracobama2181

    A Gauge Invariance of the Schrodinger Equation

    Given the schrodinger equation of the form $$-i\hbar\frac{\partial \psi}{\partial t}=-\frac{1}{2m}(-i\hbar \nabla -\frac{q}{c}A)^2+q\phi$$ I can plug in the transformations $$A'=A-\nabla \lambda$$ , $$\phi'=\phi-\frac{\partial \lambda}{\partial t}$$, $$\psi'=e^{-\frac{iq\lambda}{\hbar c}}\psi$$...
  47. Cocoleia

    Infinite square well, dimensionless Hamiltonian..

    I have always seen this problem formulated in a well that goes from 0 to L I am confused how to use this boundary, as well as unsure of what a dimensionless hamiltonian is. This is as far as I have gotten
  48. Drone0

    Electron in a triangular quantum well with triangular barrier

    Hi, it's been so long since I learned quantum mechanics. So the only thing I can solve now is the square quantum well problem. But I need help because I have to solve this problem of quantum well. I tried some calculation but not far.I try to draw the capacitance-voltage profile by drawing the...
  49. P

    Solutions to schrodinger equation with potential V(x)=V(-x)

    C is just the constant by ##\psi''## My initial attempt was to write out the schrodinger equation in the case that x>0 and x<0, so that $$ \frac {\psi'' (x)} {\psi (x)} = C(E-V(x))$$ and $$ \frac {\psi'' (-x)} {\psi (-x)} = C(E-V(-x))$$ And since V(-x) = V(x) I equated them and...