Schrodinger's equation for the complex conjugate waves function

Derivation:

The time dependent Schrodinger quation for the wave function function $psi(x,y,z,t)$ is

$-\frac{\hbar^{2}}{2m}\nabla^{2} \psi + V\psi=i\hbar\frac{\partial \psi}{\partial t} \qquad(1)$

Since wave function, $\psi$ is complex quantity i.e.

$\psi=\psi_{1}+i \: \psi_{2} \qquad(2)$

Where $\psi_{1}$ and $\psi_{2}$ are real functions of $x,y,z,t$. Substituting this form for $\psi$ in equation $(1)$, we get

$-\frac{\hbar^{2}}{2m}\nabla^{2} \left( \psi_{1}+i \: \psi_{2} \right) + V\left( \psi_{1}+i \: \psi_{2} \right) \\ \qquad = i\hbar\frac{\partial }{\partial t} \left( \psi_{1}+i \: \psi_{2} \right)$

Equation real and imaginary parts on either side of this equation, we obtain the following two equations:

$-\frac{\hbar^{2}}{2m}\nabla^{2} \psi_{1} + V\psi_{1}=-\hbar\frac{\partial \psi_{2}}{\partial t} \qquad(3)$

$-\frac{\hbar^{2}}{2m}\nabla^{2} \psi_{2} + V\psi_{2}=\hbar\frac{\partial \psi_{1}}{\partial t} \qquad(4)$

Mutiplying equation $(4)$ by $-i$ and adding it to equation $(3)$, we get

$-\frac{\hbar^{2}}{2m}\nabla^{2} \left( \psi_{1} - i \: \psi_{2} \right) + V\left( \psi_{1} - i \: \psi_{2} \right) \\ \qquad = -i\hbar\frac{\partial }{\partial t} \left( \psi_{1} - i \: \psi_{2} \right) \qquad(5)$

The complex conjugate of wave function $\psi^{*}$ is

$\psi^{*}=\psi^{*}_{1} - i \: \psi^{*}_{2} \qquad(6)$

Therefore, The equation $(5)$ can be written as

$-\frac{\hbar^{2}}{2m}\nabla^{2} \psi^{*} + V \psi^{*} =-i \hbar \frac{\partial \psi^{*}}{\partial t}$

This is the equation for complex conjugate wave function $\psi^{*}$.

Popular Posts

Study-Material













  • Classical world and Quantum world

  • Inadequacy of classical mechanics

  • Drawbacks of Old Quantum Theory

  • Bohr's Quantization Condition

  • Energy distribution spectrum of black body radiation

  • Energy distribution laws of black body radiation

  • The Compton Effect | Experiment Setup | Theory | Theoretical Expression | Limitation | Recoil Electron

  • Davisson and Germer's Experiment and Verification of the de-Broglie Relation

  • Significance of Compton's Effect

  • Assumptions of Planck’s Radiation Law

  • Derivation of Planck's Radiation Law

  • de-Broglie Concept of Matter wave

  • Definition and derivation of the phase velocity and group velocity of wave

  • Relation between group velocity and phase velocity ($V_{g}=V_{p}-\lambda \frac{dV_{p}}{d\lambda }$)

  • Group velocity is equal to particle velocity($V_{g}=v$)

  • Product of phase velocity and group velocity is equal to square of speed of light ($V_{p}.V_{g}=c^{2}$)

  • Heisenberg uncertainty principle

  • Generation of wave function for a free particle

  • Physical interpretation of the wave function

  • Derivation of time dependent Schrodinger wave equation

  • Derivation of time independent Schrodinger wave equation

  • Eigen Function, Eigen Values and Eigen Vectors

  • Postulate of wave mechanics or Quantum Mechanics

  • Quantum Mechanical Operators

  • Normalized and Orthogonal wave function

  • Particle in one dimensional box (Infinite Potential Well)

  • Minimum Energy Or Zero Point Energy of a Particle in an one dimensional potential box or Infinite Well

  • Normalization of the wave function of a particle in one dimension box or infinite potential well

  • Orthogonality of the wave functions of a particle in one dimension box or infinite potential well

  • Eigen value of the momentum of a particle in one dimension box or infinite potential well

  • Schrodinger's equation for the complex conjugate waves function

  • Probability Current Density for a free particle in Quantum Mechanics

  • Ehrenfest's Theorem and Derivation

  • Momentum wave function for a free particle

  • Wave function of a particle in free state

  • One dimensional Step Potential Barrier for a Particle

























  • Blog Archive