Schoodinger
wave equation
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chrodinger wave equation or just Schrodinger equation is one of the most fundamental equations of quantum physics and an important topic for JEE. The equation also called the Schrodinger equation is basically a differential equation and widely used in Chemistry and Physics to solve problems based on the atomic structure of matter.
Schrodinger wave equation describes the behaviour of a particle in a field of force or the change of a physical quantity over time. Erwin Schrödinger who developed the equation was even awarded the Nobel Prize in 1933.
Table of Content
What is Schrodinger Wave Equation?
Schrodinger Wave Equation Derivation
Broglie’s Hypothesis of Matter Wave
Conservation of Energy
Important Questions For Schrodinger Equation
What is Schrodinger Wave Equation?
Schrodinger wave equation is a mathematical expression describing the energy and position of the electron in space and time, taking into account the matter wave nature of the electron inside an atom.
It is based on three considerations. They are;
Classical plane wave equation,
Broglie’s Hypothesis of matter-wave, and
Conservation of Energy.
Schrodinger equation gives us a detailed account of the form of the wave functions or probability waves that control the motion of some smaller particles. The equation also describes how these waves are influenced by external factors. Moreover, the equation makes use of the energy conservation concept that offers details about the behaviour of an electron that is attached to the nucleus.
Besides, by calculating the Schrödinger equation we obtain Ψ and Ψ2, which helps us determine the quantum numbers as well as the orientations and the shape of orbitals where electrons are found in a molecule or an atom.
There are two equations, which are time-dependent Schrödinger equation and a time-independent Schrödinger equation.
Time-dependent Schrödinger equation is represented as;
i \hbar \frac{d}{d t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangleiℏdtd∣Ψ(t)⟩=H^∣Ψ(t)⟩
OR
Time-dependent Schrödinger equation in position basis is given as;
i \hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi}{\partial x^{2}}+V(x) \Psi(x, t) \equiv \tilde{H} \Psi(x, t)iℏ∂t∂Ψ=−2mℏ2∂x2∂2Ψ+V(x)Ψ(x,t)≡H~Ψ(x,t)
Where,
i = imaginary unit, Ψ = time-dependent wavefunction, h2 is h-bar, V(x) = potential and \hat{H}H^ = Hamiltonian operator.
Also Read: Quantum Mechanical Model of Atom
Time-independent Schrödinger equation in compressed form can be expressed as;

OR
Time-independent-Schrödinger-nonrelativistic-equation
\left[\frac{-\hbar^{2}}{2 m} \nabla^{2}+V(\mathbf{r})\right] \Psi(\mathbf{r})=E \Psi(\mathbf{r})[2m−ℏ2∇2+V(r)]Ψ(r)=EΨ(r)
Schrodinger Wave Equation Derivation
Classical Plane Wave Equation
A wave is a disturbance of a physical quantity undergoing simple harmonic motion or oscillations about its place. The disturbance gets passed on to its neighbours in a sinusoidal form.

The equation for the wave is a second-order partial differential equation of a scalar variable in terms of one or more space variable and time variable. The one-dimensional wave equation is-
{{\nabla }^{2}}\psi =\left( \frac{{{\vartheta }^{2}}\psi }{\vartheta {{x}^{2}}}+\frac{{{\vartheta }^{2}}\psi }{\vartheta {{y}^{2}}}+\frac{{{\vartheta }^{2}}\psi }{\vartheta {{z}^{2}}} \right)∇2ψ=(ϑx2ϑ2ψ+ϑy2ϑ2ψ+ϑz2ϑ2ψ)
The amplitude (y) for example of a plane progressive sinusoidal wave is given by:
y = A cos \left( \frac{2\pi }{\lambda }\times -\frac{2\pi t}{T}+\varphi \right),(λ2π×−T2πt+φ),
where, A is the maximum amplitude, T is the period and φ is the phase difference of the wave if any and t is the time in seconds. For a standing wave, there is no phase difference, so that,
y = A cos \left( \frac{2\pi }{\lambda }\times -\frac{2\pi t}{T} \right)(λ2π×−T2πt)= A cos \left( \frac{2\pi x}{\lambda }-2\pi vt \right),(λ2πx−2πvt), Because, v=\frac{1}{T}v=T1
In general the same equation can be written in the form of,
y={{e}^{i\left( \frac{2\pi x}{\lambda }-2\pi vt \right)}}={{e}^{-i\left( 2\pi vt-\frac{2\pi x}{\lambda } \right)}}y=ei(λ2πx−2πvt)=e−i(2πvt−λ2πx)
Broglie’s Hypothesis of Matter Wave
Planck’s quantum theory, states the energy of waves are quantized such that E = hν = 2πħν,
where, h=\frac{h}{2\pi }h=2πh and v=\frac{E}{2\pi h}v=2πhE
Smallest particles exhibit dual nature of particle and wave. De Broglie related the momentum of the particle and wavelength of the corresponding wave as follows-
\lambda =\frac{h}{mv}:λ=mvh:
where, h is Planck’s constant, m is the mass and v is the velocity of the particle.
De Broglie relation can be written as -\lambda \frac{2\pi h}{mv}=\frac{2\pi h}{p};−λmv2πh=p2πh;
where, p is the momentum.
Electron as a particle-wave, moving in one single plane with total energy E, has an
Amplitude = Wave function = Ψ ={{e}^{-i\left( 2\pi vt-\frac{2\pi x}{\lambda } \right)}}