Schrodinger equation Definition and 161 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. 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...
  2. 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 =...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. C

    I 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...
  8. 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...
  9. 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...
  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. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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] =...
  16. 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...
  17. Boltzman Oscillation

    Given this potential symmetry, how can I split up the 3D Schrodinger Equation?

    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...
  18. 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
  19. 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...
  20. 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...
  21. F

    I Hydrogen atom states

    Hello, I have a little problem understanding the quantum mechanics of a hydrogen atom. Im troubled with the following question: before i measure the state of a (simplified: without fine-, hyperfinestructure) hydrogen atom, which is the right probability density of finding the electron? is it...
  22. A

    A Do we need stochasticity in a discrete spacetime?

    Suppose that the spacetime is discrete, with only certain positions being possible for any particle. In this case, the probability distributions of particles have nonzero values at the points on which the wavefunction is defined. Do we need randomness in the transitions of particles in such a...
  23. Auto-Didact

    A Schrödinger Evolution of Self-Gravitating Disks

    This paper was recently published in the Monthly Notices of the Royal Astronomical Society. Batygin 2018, Schrödinger Evolution of Self-Gravitating Disks I am posting this in here, but I am actually more interested in the implications of looking at this the other way around: namely, from a...
  24. Mason Smith

    Finding the 10 lowest energy levels

    Homework Statement Homework Equations The Attempt at a Solution I understand the equation, and I understand the concept. My question is this: What is the best way to go about solving this problem? My line of reasoning concludes that the fourth lowest energy level is E211. However, the...
  25. Baynie

    MATLAB Code: Stationary Schrodinger EQ, E Spec, Eigenvalues

    Hello everyone, For weeks I have been struggling with this quantum mechanics homework involving writing a code to determine the energy spectrum and eigenvalues for the stationary Schrodinger equation for the harmonic oscillator. I can't find any resources anywhere. If anyone could help me get...
  26. Konte

    I About the Schrödinger equation

    Hi everybody, For solving the time dependent Schrödinger equation ##H|\psi(t) \rangle = i\hbar \frac{\partial}{\partial t}|\psi (t)\rangle##, I read in quantum mechanics books the assumption about the solution ##|\psi(t) \rangle## which is made of a linear combination of a complete set of the...
  27. S

    Prove that ##\psi## is a solution to Schrödinger equation

    Homework Statement For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the...
  28. D

    Finding Stationary Wavefunction with a Line Potential

    Homework Statement A particle of mass m in one dimension has a potential: $$V(x) = \begin{cases} V_0 & x > 0 \\ 0 & x \leq 0 \end{cases} $$ Find ##\psi(x)## for energies ##0 < E < V_0##, with parameters $$k^2 = \frac{2mE}{\hbar^2}$$ and $$\kappa^2 = \frac{2m(V_0 - E)}{\hbar^2}$$...
  29. Rupul Chandna

    I Why is separation constant l(l+1) instead of +-l^2?

    While separating variables in the Schrodinger Equation for hydrogen atom, why are we taking separation constant to be l(l+1) instead of just l^2 or -l^2, is it just to make the angular equation in the form of Associated Legendre Equation or is there a deeper meaning to it?
  30. Matt Chu

    Time Derivative of Expectation Value of Position

    Homework Statement I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##. Homework Equations $$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$ The Attempt at a Solution [/B] So...
  31. P

    Finding the quantized energies of a particle

    Homework Statement Okay, so the question i'm trying to solve is to find the quantized energies for a particle in the potential: $$V(x)=V_0 \left ( \frac{b}{x}-\frac{x}{b} \right )^2$$ for some constant b. The Attempt at a Solution I am following along with the derivation of the quantized...
  32. J

    Quantum I need a book to solve Schrodinger's eqn numerically

    I have found this one that looks perfect: https://www.amazon.com/dp/331999929X/?tag=pfamazon01-20 THe problem is that it has not been published yet :( , but I can't believe there is no other book on the subject. What I want is to solve numerically the Schrodinger equation with no special...
  33. Ziezi

    Discrepancies between numerical and analytical solutions

    The analytical solutions are: \begin{equation} \psi(x) = \begin{cases} Ce^{\alpha x}, \text{if } x < -\frac{L}{2}\\ Asin(kx) + Bcos(kx), \text{if } -\frac{L}{2} \leq x \leq \frac{L}{2}\\ Fe^{-\alpha x} , \text{if } x > \frac{L}{2} \end{cases} \end{equation}...
  34. A

    I Finite square well bound states

    Let's suppose I have a finite potential well: $$ V(x)= \begin{cases} \infty,\quad x<0\\ 0,\quad 0<x<a\\ V_o,\quad x>a. \end{cases} $$ I solved the time-independent Schrodinger equation for each region and after applying the continuity conditions of ##\Psi## and its derivative I ended up with...
  35. 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.
  36. It's me

    Dirac hydrogen atom vs spin symmetry

    Homework Statement Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and...
  37. Safder Aree

    Normalization of wave function

    Homework Statement I have the wave function Ae^(ikx)*cos(pix/L) defined at -L/2 <= x <= L/2. and 0 for all other x. The question is: A proton is in a time-independent one-dimensional potential well.What is the probability that the proton is located between x = − L/4 and x = L/4 ? Homework...
  38. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 1: Black Body Radiation I

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 1: Black Body Radiation I

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  39. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 2: Black Body Radiation II

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 2: Black Body Radiation II

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  40. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 3: Black Body Radiation III

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 3: Black Body Radiation III

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  41. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 6: Black Body Radiation VI

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 6: Black Body Radiation VI

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  42. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 7: Black Body Radiation VII

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 7: Black Body Radiation VII

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  43. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 8: Radiation as a collection of particles called photons

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 8: Radiation as a collection of particles called photons

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  44. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 9: Quantum Hypothesis and specific heat of solids

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 9: Quantum Hypothesis and specific heat of solids

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  45. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 10: Bohr's Model of hydrogen spectrum

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 10: Bohr's Model of hydrogen spectrum

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  46. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 11: Wilson Sommerfeld quantum condition I

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 11: Wilson Sommerfeld quantum condition I

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  47. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 12: Wilson Sommerfeld quantum condition II

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 12: Wilson Sommerfeld quantum condition II

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  48. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 13: Wilson Sommerfeld quantum condition III

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 13: Wilson Sommerfeld quantum condition III

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  49. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 14: Quantum conditions and atomic structure

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 14: Quantum conditions and atomic structure

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
  50. Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 15: Eienstien's A and B coefficients

    Introductory Quantum Mechanics with Prof. Manoj Harbola (NPTEL):- Lecture 15: Eienstien's A and B coefficients

    All copyright reserved to Prof. Harbola and NPTEL, Govt. of India. Duplication punishable offence. Course Website: http://www.nptel.ac.in/courses/115104096/
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