SCHRODINGER WAVE EQUATION

April 15, 2024
by Bhoomika Sheladiya

In 1926, Erwin Schrodinger gave a wave equation to explain the behaviour of electron waves in atoms and molecules. In the Schrodinger wave model of an atom, the discrete energy levels or atoms are proposed by Bohar.

Consider simple wave motion as that of the vibration of a stretched string. Suppose y be the amplitude of this vibration at the equation for such wave motion may be expressed,

\[\displaystyle \frac{{{{\partial }^{2}}\psi }}{{\partial {{x}^{2}}}}=\frac{1}{{{{u}^{2}}}}\times \frac{{{{\partial }^{2}}y}}{{\partial {{t}^{2}}}}---(1)\]

Here, u = velocity with the wave propagating. Two variables x and t in the above differential equation, the amplitude y depends upon two variables x and t.

To solve differential equation (1), it is necessary to separate the two variables. So, y may be expressed as,

\[\displaystyle y=f(x)g(t)---(2)\]

Here, f(x) = function of the coordinate x only and , g(t) =function of the time t only

For stationary waves, like stretched string, the function g(t) represented the expression,

\[\displaystyle g(t)=A\sin (2\pi \nu t)----(3)\]

Here, v = vibrational frequency

A = constant known as the maximum amplitude

For stationary waves, the equation for y may be written as,

\[\displaystyle y=f(x)A\sin (2\pi \nu t)----(4)\]

So,

\[\displaystyle \frac{{{{\partial }^{2}}y}}{{\partial {{t}^{2}}}}=-f(x)4{{\pi }^{2}}{{\nu }^{2}}A\sin (2\pi \nu t)---(5)\]
\[\displaystyle =-4{{\pi }^{2}}{{\nu }^{2}}f(x)g(t)---(6)\]

Similarly, follows from equation (4) that,

\[\displaystyle \frac{{{{\partial }^{2}}y}}{{\partial {{x}^{2}}}}=\frac{{{{\partial }^{2}}f(x)}}{{\partial {{x}^{2}}}}g(t)----(7)\]

Combining equations (1), (6), and (7), we get,

\[\displaystyle \frac{{{{\partial }^{2}}f(x)}}{{\partial {{x}^{2}}}}=-\frac{{4{{\pi }^{2}}{{\nu }^{2}}}}{{{{\upsilon }^{2}}}}f(x)---(8)\]

The frequency of the vibration v is related to the velocity u by the expression u = νλ; here, λ = corresponding wavelength; from equation (8), we get,

\[\displaystyle \frac{{{{\partial }^{2}}f(x)}}{{\partial {{x}^{2}}}}=-\frac{{4{{\pi }^{2}}}}{{{{\lambda }^{2}}}}f(x)---(9)\]

Equation (9) is only valid for the wave motion in one dimension. Extend it to three dimensions represented by the coordinates x, y and z.

Evidently, f(x) is replaced by the amplitude function for the three coordinates, Ψ(x, y, z). For the sake of simplicity, it may be put merely as Ψ taken from equation (9)

\[\displaystyle \frac{{{{\partial }^{2}}\psi }}{{\partial {{x}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{y}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{z}^{2}}}}=-\frac{{4{{\pi }^{2}}}}{{{{\lambda }^{2}}}}\psi ---(10)\]

Following de Broglie’s ideas, the Austrian Physicist Schrodinger applied the above treatment to material waves associated with all particles, including electrons, atoms and molecules.

Incorporating de Broglie’s relationship, λ = h/mu, in equation (10) we have,

\[\displaystyle \frac{{{{\partial }^{2}}\psi }}{{\partial {{x}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{y}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{z}^{2}}}}=-\frac{{4{{\pi }^{2}}{{m}^{2}}{{u}^{2}}}}{{{{h}^{2}}}}\psi ---(11)\]

Here, m= mass, u= velocity of the particle the kinetic energy of the particle is ½ mu2, which is equal to the total energy minus the potential energy v of the particle, i.e.

\[\displaystyle K.E.=\frac{1}{2}m{{u}^{2}}=E-V\text{ or }m{{u}^{2}}=2(E-V)---(12)\]

combining this result with the equation (11) we get,

\[\displaystyle \frac{{{{\partial }^{2}}\psi }}{{\partial {{x}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{y}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{z}^{2}}}}+\frac{{8{{\pi }^{2}}m(E-V)}}{{{{h}^{2}}}}\psi =0---(13)\]

Equation (13) is called the Schrödinger equation.

It is the most celebrated equation in wave mechanics that is written as,

\[\displaystyle \left[ {-\frac{{{{\hbar }^{2}}}}{{2m}}\left( {\frac{{{{\partial }^{2}}\psi }}{{\partial {{x}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{y}^{2}}}}+\frac{{{{\partial }^{2}}\psi }}{{\partial {{z}^{2}}}}} \right)+V} \right]=E\psi ---(14)\]

Or

\[\displaystyle \left[ {-\frac{{{{\hbar }^{2}}}}{{2m}}{{\nabla }^{2}}+V} \right]\psi =E\psi \left( {\hbar =\frac{h}{{2\pi }}} \right)----(15)\]

Here,

\[\displaystyle {{\nabla }^{2}}=\text{Laplacin operator,}\]
\[\displaystyle {{\nabla }^{2}}=\frac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}}+\frac{{{{\partial }^{2}}}}{{\partial {{y}^{2}}}}+\frac{{{{\partial }^{2}}}}{{\partial {{z}^{2}}}}---(16)\]

Defining the Hamiltonian operator Ĥ as,

\[\displaystyle \hat{{\mathrm H}}=-\frac{{{{\hbar }^{2}}}}{{2m}}{{\nabla }^{2}}+V----(17)\]

The time-dependent Schrodinger equation (15) becomes,

\[\displaystyle \hat{{\mathrm H}}\psi =E\psi ---(18)\]
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About the author

Bhoomika Sheladiya

BSc. (CHEMISTRY) 2014- Gujarat University
MSc. (PHYSICAL CHEMISTRY) 2016 - School of Science, Gujarat University

Junior Research Fellow (JRF)- 2019
AD_HOC Assistant Professor-(July 2016 to November 2021)

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