# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

43. ### 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...
44. ### 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.>
45. ### 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 ?
46. ### 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...
47. ### 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 $$A'=A-\nabla \lambda$$ , $$\phi'=\phi-\frac{\partial \lambda}{\partial t}$$, $$\psi'=e^{-\frac{iq\lambda}{\hbar c}}\psi$$...
48. ### 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
49. ### 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...
50. ### 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...