Bound State Solutions of the Schrödinger Equation

  • Ajoy Ghatak
  • S. Lokanathan
Part of the Fundamental Theories of Physics book series (FTPH, volume 137)

Abstract

When the Hamiltonian for a system is independent of time, there is an essential simplification in that the general solution of the Schrödinger equation can be expressed as a function of spatial coordinates and a function of time. Thus, assuming the potential energy function to be independent of time, the one-dimensional time dependent Schrödinger equation [see Eq. (25) of Chapter 4] is given by
$$ i\frac{{\partial \psi }}{{\partial \mu }} = - \frac{{{^2}}}{{2\mu }}\,\frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + V\left( x \right)\;\psi \left( {x,t} \right) $$
(1)
where μ represents the mass of the particle. The above equation can be solved by using the method of separation of variables
$$ \psi \left( {x,t} \right) = \psi \left( x \right)\;T\;\left( t \right) $$
(2)
Substituting in Eq. (1) and dividing by ψ(x, t), we obtain
$$ \frac{{i}}{{T\left( t \right)}}\;\frac{{dT}}{{dt}} = \frac{1}{{\psi \left( x \right)}}\left[ { - \frac{{{^2}}}{{2\mu }}\;\frac{{{d^2}\psi }}{{d{x^2}}} + V\left( x \right)\;\psi \;\left( x \right)} \right] $$
(3)

Keywords

Peaked 

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References

  1. 1.
    D. Halliday and R. Resnick, Physics Parts I & II, John Wiley, New York (1978).Google Scholar
  2. 2.
    S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover Publications, New York (1957).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2004

Authors and Affiliations

  • Ajoy Ghatak
    • 1
  • S. Lokanathan
    • 2
  1. 1.Indian Institute of TechnologyNew DelhiIndia
  2. 2.Jawahar Lal Nehru PlanetariumBangaloreIndia

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