Schrodinger Wave Equation

Schrodinger 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 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;

iddtΨ(t)=H^Ψ(t)i \hbar \frac{d}{d t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle

OR

Time-dependent Schrödinger equation in position basis is given as;

iΨt=22m2Ψx2+V(x)Ψ(x,t)H~Ψ(x,t)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)

Where,

i = imaginary unit, Ψ = time-dependent wavefunction, h2 is h-bar, V(x) = potential and H^\hat{H} = Hamiltonian operator.

Also Read: Quantum Mechanical Model of Atom

Time-independent Schrödinger equation in compressed form can be expressed as;

Time-independent Schrödinger equation

OR

Time-independent-Schrödinger-nonrelativistic-equation

[22m2+V(r)]Ψ(r)=EΨ(r)\left[\frac{-\hbar^{2}}{2 m} \nabla^{2}+V(\mathbf{r})\right] \Psi(\mathbf{r})=E \Psi(\mathbf{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.

Classical Plane Wave Equation

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-

2ψ=(ϑ2ψϑx2+ϑ2ψϑy2+ϑ2ψϑz2){{\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)

The amplitude (y) for example of a plane progressive sinusoidal wave is given by:

y = A cos (2πλ×2πtT+φ),\left( \frac{2\pi }{\lambda }\times -\frac{2\pi t}{T}+\varphi \right),

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 (2πλ×2πtT)\left( \frac{2\pi }{\lambda }\times -\frac{2\pi t}{T} \right)= A cos (2πxλ2πvt),\left( \frac{2\pi x}{\lambda }-2\pi vt \right), Because, v=1Tv=\frac{1}{T}

In general the same equation can be written in the form of,

y=ei(2πxλ2πvt)=ei(2πvt2πxλ)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)}}

Broglie’s Hypothesis of Matter Wave

Planck’s quantum theory, states the energy of waves are quantized such that E = hν = 2πħν,

where, h=h2πh=\frac{h}{2\pi } and v=E2πhv=\frac{E}{2\pi h}

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-

λ=hmv:\lambda =\frac{h}{mv}:

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 λ2πhmv=2πhp;-\lambda \frac{2\pi h}{mv}=\frac{2\pi h}{p};

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 =ei(2πvt2πxλ)={{e}^{-i\left( 2\pi vt-\frac{2\pi x}{\lambda } \right)}}

Substituting for wavelength and energy in this equation,

Amplitude = Wave function = Ψ =ei(2πEt2πh2πpx2πh)=eih(Etpx)={{e}^{-i\left( \frac{2\pi Et}{2\pi h}-\frac{2\pi px}{2\pi h} \right)}}={{e}^{-\frac{i}{h}\left( Et-px \right)}}

Now partial differentiating with respect to x, ϑ2ψϑx2=p2h2ψ\frac{{{\vartheta }^{2}}\psi }{\vartheta {{x}^{2}}}=\frac{{{p}^{2}}}{{{h}^{2}}}\psi OR p2ψ=h2ϑ2ψϑx2{{p}^{2}}\psi =-{{h}^{2}}\frac{{{\vartheta }^{2}}\psi }{\vartheta {{x}^{2}}}

Also partial differentiating with respect to t, ϑψϑt=iEhψ\frac{\vartheta \psi }{\vartheta t}=-\frac{iE}{h}\psi OR Eψ=hiϑψϑt=ihϑψϑtE\psi =-\frac{h}{i}\frac{\vartheta \psi }{\vartheta t}=ih\frac{\vartheta \psi }{\vartheta t}

Conservation of Energy

Total energy is the sum of the kinetic and potential energy of the particle.

E = KE + PE =mv22+U=p22m+=\frac{m{{v}^{2}}}{2}+U=\frac{{{p}^{2}}}{2m}+U: p = mv

Substituting in the wave function equation,

Eψ=ψp22m+UψE\psi =\frac{\psi {{p}^{2}}}{2m}+U\psi

Substituting for EΨ and p2Ψ, we get the wave function for one-dimensional wave called as “Time-dependent Schrodinger wave equation”.

Eψ=ihϑψϑt=h2ϑ2ψ2mϑx2+UψE\psi =i\,\,h\frac{\vartheta \psi }{\vartheta t}=-\frac{{{h}^{2}}{{\vartheta }^{2}}\psi }{2m\,\vartheta {{x}^{2}}}+U\psi

Time dependent Schrodinger equation for three-dimensional progressive wave then is,

Eψ=h22m(ϑ2ψϑx2+ϑ2ψϑy2+ϑ2ψϑz2)+UψE\psi =-\frac{{{h}^{2}}}{2m}\left( \frac{{{\vartheta }^{2}}\psi }{\vartheta {{x}^{2}}}+\frac{{{\vartheta }^{2}}\psi }{\vartheta {{y}^{2}}}+\frac{{{\vartheta }^{2}}\psi }{\vartheta {{z}^{2}}} \right)+U\psi

On rearranging,

(ϑ2ψϑx2+ϑ2ψϑy2+ϑ2ψϑz2)+2mh2(EU)ψ=0\left( \frac{{{\vartheta }^{2}}\psi }{\vartheta {{x}^{2}}}+\frac{{{\vartheta }^{2}}\psi }{\vartheta {{y}^{2}}}+\frac{{{\vartheta }^{2}}\psi }{\vartheta {{z}^{2}}} \right)+\frac{2m}{{{h}^{2}}}\left( E-U \right)\psi =0

The equation is also written as-

2ψ+2mh2(EU)ψ=0{{\nabla }^{2}}\psi +\frac{2m}{{{h}^{2}}}(E-U)\psi =0 where, 2ψ=(ϑ2ψϑx2+ϑ2ψϑy2+ϑ2ψϑz2){{\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)

Schrodinger equation is written as HΨ = EΨ, where h is said to be a Hamiltonian operator.

Important Questions For Schrodinger Equation

1. What is a wave function?

Answer: Wave function is used to describe ‘matter waves’. Matter waves are very small particles in motion having a wave nature – dual nature of particle and wave. Any variable property that makes up the matter waves is a wave function of the matter-wave. Wave function is denoted by a symbol ‘Ψ’.

Amplitude, a property of a wave, is measured by following the movement of the particle with its Cartesian coordinates with respect of time. The amplitude of a wave is a wave function. The wave nature and the amplitudes are a function of coordinates and time, such that,

Wave function Amplitude = Ψ = Ψ(r,t); where, ‘r’ is the position of the particle in terms of x, y, z directions.

2. What is meant by stationary state and what is its relevance to atom?

Answer: Stationary state is a state of a system, whose probability density given by | Ψ2 | is invariant with time. In an atom, the electron is a matter wave, with quantized angular momentum, energy, etc. Movement of the electrons in their orbit is such that probability density varies only with respect to the radius and angles.

The movement is akin to a stationary wave between two fixed ends and independent of time. Wave function concept of matter waves are applied to the electrons of an atom to determine its variable properties.

3. What is the physical significance of Schrodinger wave function?

Answer: Bohr concept of an atom is simple. But it cannot explain the presence of multiple orbitals and the fine spectrum arising out of them. It is applicable only to the one-electron system.

Schrodinger wave function has multiple unique solutions representing characteristic radius, energy, amplitude. Probability density of the electron calculated from the wave function shows multiple orbitals with unique energy and distribution in space.

Schrodinger equation could explain the presence of multiple orbitals and the fine spectrum arising out of all atoms, not necessarily hydrogen-like atoms.

4. What is the Hamilton operator used in the Schrodinger equation?

Answer: In mathematics, the operator is a rule, that converts observed properties into another property. For example, ‘A’ will be an operator if it can change a property f(x) into another f(y). f(x)= f(y) Hamiltonian operator is the sum of potential and kinetic energies of particles calculated over three coordinates and time.

Hamiltonian operator = Ȟ = T + V = Kinetic energy + Potential energy

Ȟ = h22m()2-\frac{{{h}^{2}}}{2m}{{(\nabla )}^{2}} + V( r,t)

5. The electrons are more likely to be found:

Electron Region

(1) in the region a and b

(2) in the region a and c

(3) only in the region c

(4) only in the region a

Region a and c has the maximum amplitude (Ψ) and hence the maximum probability density of Electrons | Ψ2 |
Answer: (2)

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