What is Schrodinger's equation: Definition and 106 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. MevsEinstein

    B Particle Function? Particle Equation?

    The following is the wave equation from Electrodynamics: $$\frac{\partial^2 \Psi}{\partial t^2} = c^2\frac{\partial^2 \Psi}{\partial x^2}$$ Where ##\Psi## is the wave function. But because of Heisenberg's Uncertainty, physicists had to come up with another equation (the Schrodinger equation)...
  2. this_is_my_name

    Calculate qubit states with Schrodinger's equation

    Summary:: How to calculate qubit states with the Schrodinger eq I'm writing something about the relation between quantum computers and the Schrodinger equation. One of the requirements is there has to be an experiment. So I thought I could measure some qubits that have results and then do the...
  3. G

    I Conventional description of the matter wave

    I have been working on a relatively simple problem. Just take a quantum wave function for which a physical requirement is that an arbitrary displacement of x or an arbitrary shift of t should not alter the character of the wave, and I want to find the state function solution. A possible guess...
  4. Jarvis323

    A Interpreting QM without Schrödinger's equation

    Rovelli, in his recent paper, writes: https://arxiv.org/pdf/2109.09170.pdf Do you think Rovelli is correct in implying the wave function is redundant and misleading? If we disregard the wave function's significance, beyond convenience, say we forgot it even existed, how would that change (or...
  5. 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...
  6. P

    A Solving Schrödinger's Equation with a Smooth Potential Wall: A Detailed Guide

    Hello everyone, I'm looking for help for the problem 3 of the chapter III. Schrödinger's equation, §25 The transmission coefficient of the Volume 3 of the Landau-Lifshitz book (non-relativistic QM). In this exercise Landau considers a smooth potential wall $$\frac{U_0}{1 + \exp{\left(-\alpha x...
  7. 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...
  8. A

    I How to derive Schrodinger's equation?

    Homework Statement:: Derive Schrodinger equation Relevant Equations:: Schrodinger Equation I want to find the derivation of Schrodinger Equation. Actually, I learned quantum mechanics already, but I think the proof that begins from the plane wave solution is quit ambiguous. Because I feel...
  9. eloiseh

    Find potential energy using time-independent Schrodinger's equation

    I had found what U(x) was equal to already by plugging in the wave function and simplifying, which is (2h^2/mL^4)(x^2 - 3L^2/2) by the way. But the solution key that I have goes an extra step. After stating the equation of U(x) that I got, it says that: "U(x) is a parabola centred at x = 0 with...
  10. 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...
  11. L

    I Why was Schrodinger's equation for the relativistic wave unsuccessful?

    why was the schrodinger equation for the relativistic wave unsuccessful?
  12. currently

    Schrodinger's Equation in terms of vacuum permittivity?

    Summary:: How can Schrodinger's Equation be written relative to vacuum permittivity I am wondering why a particular problem uses this equation: It is stated to be Schrodinger's equation. Where does the potential come in, as well as the e^2/r ? An explanation would be greatly appreciated. Thanks.
  13. 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...
  14. J

    Help Verifying Schrodinger's equation in different potentials

    There are 3 regions, to which I split the function as follows. I can derive the solutions myself. However I need to verify whether I am using them properly. There are two principles/ideas that I am not sure if I am misinterpreting. 1) Anytime a wave is incident on a discontinuity(such as when a...
  15. fluidistic

    A Schrödinger's equation: a diffusion or a wave equation?

    From many sources (Internet, Landau & Lifshitz, etc.), it is claimed that the Schrödinger's equation is a wave equation. However I do not understand why for the following reasons: It is Galilean invariant, unlike the wave equation which is Lorentz invariant. Note that the diffusion/heat...
  16. G

    A Seeking a derivation of Schrödinger's wave equation

    I am interested in the derivation of Schrödinger’s wave equation from the Klein Gordon equation. I have looked in Penfold’s ‘The Road to Reality’, the open University’s Quantum Mechanics books, Feynman’s lectures, the internet, but not found what I want. Everyone seems to take it as a given...
  17. S

    Radial Distribution Function (RDF) versus Radial component

    Homework Statement For the RDF, we take the square of the radial component multiplied by 4pi r2 (the surface area of a sphere) and this gives us the probability density of finding an electron r distance away. Whats the point in multiplying it by r2? Homework Equations RDF= r2[R(r)]2 The...
  18. K

    I Difference between Schrodinger's equation and wave function?

    Is there a difference between Schrodinger's equation and the wave function? In the beginning of the second edition by David J. Griffiths he compares the classical F(x,t) and Schrodinger's equation and I am having trouble understanding the connection.
  19. A

    Solving Schrodinger's Equation with a weak Imaginary Potential

    Homework Statement A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E. a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1. b) Calculate the probability current density...
  20. C

    (Numerical) Boundary Value Problem for Schrodinger's Equation

    Homework Statement Suppose we have the standard rectangular potential barrier in 1D, with $$ V = \left\{ \! \begin{aligned} 0 & \,\text{ if } x<0, x>d\\ V_0 & \,\text{ if } x>0,x<d\\ \end{aligned} \right. $$ The standard approach to solve for tunneling through the barrier is to match the...
  21. C

    I Energy of an electron with Schrodinger's equation

    Hello, I was trying to make a simple model of an electron tunneling through several potential barriers. The electron will flow through a conductor to a heterojunction of possibly semiconductor/oxide layers. I assume the electron is coming as a plane wave from the left with some energy E. We know...
  22. Z

    I How Fermat's principle interprets diffraction and interference

    <Moved from the homework section> 1. Homework Statement I have read several chapters of De Brogile's article "the theory of quanta".The motion of a particle could be analogious to a ray in general optics.This is an analogy between Maupertui's principle and fermat's principle. How to use this...
  23. L

    I Solving Schrodinger's equation

    What are the key things to look when solving Schrodinger equation for the particular system like Hydrogen atom
  24. TheSodesa

    The width of a finite potential well

    Homework Statement An electron is enclosed in a potential well, whose walls are ##V_0 = 8.0eV## high. If the energy of the ground state is ##E = 0.50eV##, approximate the width of the well. Answer: ##0.72nm## Homework Equations For an electron in a potential well, whose energy is less than...
  25. B

    I Schrödinger's Equation Infinite Potential Well

    Given the equation ##\frac{d^2 \psi (x)}{{dt}^2}+\frac{2m}{{\hbar}^2}(E-V(x))=0## the general solution is: $$\psi (x)=A_1 e^{ix \sqrt{\frac{2m}{{\hbar}^2}(E-V(x))}} +A_2 e^{-ix \sqrt{\frac{2m}{{\hbar}^2}(E-V(x))}}$$ If we have an infinite potential well: ## V(x)=\begin{cases} \infty \quad x\ge...
  26. Fuinne

    B A question about Schrödinger's equation.

    Hi, So was Schrödinger's equation basically the birth of the idea that quantum mechanical systems work off probability? Also, I'm sure it's not Heisenberg, but I'm thinking of a physicist who took the wave function Ψ(x,y,z,t) and squared the absolute value of it, and I was wondering what his...
  27. 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...
  28. S

    I Evalutaion of Schrodinger's equation

    hi! i asked to evaluate the schrodinger equation using dirac notaion. i saw some ways but didn't understand them. is it true? if it does, what are M and 1 represent? thanks!
  29. Y

    Wave Functions With Same Energies Are the Same (only differ by a complex phase)

    Homework Statement Assume a particle with a wave function ##\psi(x)## such that ##-\infty < x < \infty##, that move under some potential ##V(x)##. Show that: a) two wave functions with same energies can only differ by a complex phase; b) if the potential is real, then you can choose the wave...
  30. J

    I How to prove Schrodinger's equation in momentum space?

    Specifically, i do not know hot to express the potential in momentum space. If someone would provide me with a link of source that has the proof in it, it would be appreciated.
  31. B

    Can Schrodinger's Equation predict steric clashes

    Hi All: My name is Ben, and l am new to the forum. Currently doing a Med Degree, but already have an Engineering Degree and Masters. With all Schrodinger's equation is able to do, could it also predict and model the steric clashes of functional groups in multi domain proteins, or enzymes?
  32. H

    Derivation of Schrodinger's equation in momentum space

    Homework Statement How do you get from (3.171) to (3.172)? In particular, why is ##\int e^{-ip.r/{\hbar}}\frac{p_{op}^2}{2m}\Psi(r,t)\,dr=\int\frac{p_{op}^2}{2m}[e^{-ip.r/{\hbar}}\Psi(r,t)]\,dr##? ##\,\,\,\,\,##-- (1) Homework Equations The Attempt at a Solution For (1) to be true, it...
  33. A

    Wave function description and Schrödinger's equation

    Hello I am not professional at physics and new on this forum so don't be angry when I make mistakes So my question is about wave function so is it right that ψ=Asin(kx)+Bcos(kx) where A and B are constants, k is a some constant k=√2mE/ħ^2 and x is cordinate so when we give A and B value and do...
  34. 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??)...
  35. 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...
  36. Priyadarshini

    What is the Significance of Schrodinger's Wave Equation in Quantum Mechanics?

    Why would one need to use the Schrodinger's wave equation? What does it give us? I understand that we solve for psi and stuff, but graphically what does it give us, in practical and not theoretical terms? What exactly is a wave function? After solving for psi with the help of the equation, we...
  37. S

    Schrödinger Eq: Questions on Mass, Uncertainty & Pi

    Some questions on Schrödingers equation (se)... 1) what does the m in se refer too the equation describes a wave function not a particle so how can we associate a mass with it. 2) the equation has to have fixed numbers ie it can't have any numbers with uncertainies yet shouldn't mass be...
  38. I

    Potential energy term in Schrodinger's equation?

    consider a 1-D problem where a charged particle travels along +x with a bound state energy. the vertical potential energy axis is displaying information about the force that is parallel to the particle's axis of motion. is this correct? so then this would be an accurate representation? where...
  39. A

    Some help with quantum mechanics?

    Homework Statement Hi everyone. This is my first time on her so I hope I make what I'm looking for clear! The question in the book says, If V(x) = ∞, x<0 ; -Vo, 0 < x < a ; 0, x > a Solve the schrodinger equation for E < 0 inside and outside the well. Apply the boundary conditions at x...
  40. 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\{...
  41. P

    Deriving the statistical interpretation from Schrodinger's equation?

    So, there are two things in Quantum Mechanics that I understand are axioms: the first is the schrodinger equation, which cannot be derived. Okay fine, we have to start somewhere. The second axiom is that the integral from a to b of the wavefunction-mod-squared gives the probability of finding...
  42. SamRoss

    Where are charge and spin in Schrodinger's equation?

    Schrodinger's equation is what is used to determine the probability of finding a particle somewhere, but where are charge and spin? How do you know if the probability wave you solved for is for a proton or electron or some other particle? I see m in the equation for plugging in mass, but nothing...
  43. U

    Proper form of schrodinger's equation?

    I feel a bit silly asking this, but I've been working through some QM lately and there's one aspect of Schrodinger's equation that's puzzling me. I've typically understood the equation as i\hbar \frac{d|\psi\rangle}{dt}=\hat H |\psi\rangle, but I've also seen it written as i\hbar \frac{\partial...
  44. L

    Questions about Schrödinger's Equation: Help Needed!

    Hi everyone I have to questions which I don't have an answer to: 1. the solution for the Schrödinger's equation are continuous (for time as well as for the location). But why do I get discrete values for the energies for example (let's say in hydrogen) ? 2. Is there a spherical harmonic...
  45. fluidistic

    Solving Schrödinger's Equation for Cylindrical Boundaries

    Homework Statement I must get the first eigenvalues of the time independent Schrödinger's equation for a particle of mass m inside a cylinder of height h and radius a where ##h \sim a##. The boundary conditions are that psi is worth 0 everywhere on the surface of the cylinder. Homework...
  46. R

    Schrödinger's equation, its derivation

    Richard Feynman said: So does this mean that it is a equation like the Balmer formula or the Rydberg equation? There's no theory behind it, it's just an empirical formula? Did S just look at data and come up with an equation that fit the data? Is it possible that we will find out why the...
  47. H

    Schrodinger's equation and the finite well(conceptual)

    Hello, I have a question about Schrodinger's equation and the finite well. It isn't so much as a math question but rather how to interpret the problem. I'll use the picture on the right from here for reference and for simplicity, I'll stick to one dimension. When I think of this problem, I...
  48. B

    Unlocking Schrödinger's Equation Through Multivariable Calculus

    I went through a whole calculus book and I didn't find anything that resembled sch eq. I specifically wanted to learn about that. What do you think the chapter will be called that introduced the math necessary for sch eq? Maybe it's in multivariable calculus.
  49. E

    Rest energy in Schrodinger's equation

    Why is the rest energy usually ignored in Schrodinger’s equation? (I am aware of Dirac’s later relativistic equation.) What is the justification? Wouldn’t it change the nature of the solutions to the last equation below if it were included? Well, ok, it won't copy my Word equations. Why...