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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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...

35. 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. 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. 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

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

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

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

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

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

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

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

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

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

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

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

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

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