Question

prove Schrodinger's equation​

Answer 1

Answer:

Schrodinger Equation is a mathematical expression which describes the change of a physical quantity over time in which the quantum effects like wave-particle duality are significant. The Schrodinger Equation has two forms the time-dependent Schrodinger Equation and the time-independent Schrodinger Equation.

Answer 2

Time-dependent equation

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation (TDSE), which gives a description of a system evolving with time:[5]:143

A wave function that satisfies the nonrelativistic Schrödinger equation with V = 0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here.

Time-dependent Schrödinger equation (general)

{\displaystyle i\hbar {\frac {d}{dt}}\vert \Psi (t)\rangle ={\hat {H}}\vert \Psi (t)\rangle } {\displaystyle i\hbar {\frac {d}{dt}}\vert \Psi (t)\rangle ={\hat {H}}\vert \Psi (t)\rangle }

where {\displaystyle i} i is the imaginary unit, {\displaystyle \hbar ={\frac {h}{2\pi }}} \hbar ={\frac {h}{2\pi }} is the reduced Planck constant, {\displaystyle \Psi } \Psi (the Greek letter psi) is the state vector of the quantum system, {\displaystyle t} t is time, and {\displaystyle {\hat {H}}} {\hat {H}} is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector {\displaystyle \vert \mathbf {r} \rangle } {\displaystyle \vert \mathbf {r} \rangle }. It is a scalar function, expressed as {\displaystyle \Psi (\mathbf {r} ,t)=\langle \mathbf {r} \vert \Psi \rangle } {\displaystyle \Psi (\mathbf {r} ,t)=\langle \mathbf {r} \vert \Psi \rangle }. Similarly, the momentum-space wave function can be defined as {\displaystyle {\tilde {\Psi }}(\mathbf {p} ,t)=\langle \mathbf {p} \vert \Psi \rangle } {\displaystyle {\tilde {\Psi }}(\mathbf {p} ,t)=\langle \mathbf {p} \vert \Psi \rangle }, where {\displaystyle \vert \mathbf {p} \rangle } {\displaystyle \vert \mathbf {p} \rangle } is the momentum eigenvector.

Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".

The most famous example is the nonrelativistic Schrödinger equation for the wave function in position space {\displaystyle \Psi (\mathbf {r} ,t)} \Psi (\mathbf {r} ,t) of a single particle subject to a potential {\displaystyle V(\mathbf {r} ,t)} {\displaystyle V(\mathbf {r} ,t)}, such as that due to an electric field.[6][note 2]

Time-dependent Schrödinger equation in position basis

(single nonrelativistic particle)

{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\right]\Psi (\mathbf {r} ,t)} {\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\right]\Psi (\mathbf {r} ,t)}

where {\displaystyle m} m is the particle's mass, and {\displaystyle \nabla ^{2}} \nabla ^{2} is the Laplacian.