Time-Independent Schrödinger Equation

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SUMMARY

The time-independent Schrödinger equation (SE) is fundamentally the spectral equation for the Hamilton operator. It is derived from the general Schrödinger equation by postulating unitary evolution of physical states and considering the self-adjoint generator of symmetry. In cases where the Hamiltonian is time-independent, one can isolate the time component of the state vector, leading to the spectral equation. This equation is crucial for determining the basis of the vector space of possible physical states and the energy values of the system.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with the Hamilton operator
  • Knowledge of unitary evolution in quantum systems
  • Basic concepts of spectral theory
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  • Study the derivation of the time-independent Schrödinger equation from the general Schrödinger equation
  • Explore the properties of the Hamilton operator in quantum mechanics
  • Learn about self-adjoint operators and their role in quantum theory
  • Investigate spectral theory and its applications in quantum mechanics
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coki2000
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How can time-independent schrödinger equation be proven? Do you know any source which explains it clearly? Thanks for replies.
 
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The so-called time independent SE is nothing else but the spectral equation for the Hamilton operator. One postulates the general Schroedinger equation (alternatively one postulates a unitary evolution of physical states and then derives the SE by considering the self-adj generator of the symmetry) from which then, in the very fortunate case in which the Hamiltonian is time-independent, one can separate the time-component of the state vector completely and end up with the spectral equation of the Hamiltonian. Solving it would normally provide us the the basis for the vector space of possible physical states of the system. And the possible values for the energy of the system.
 
coki2000 said:
How can time-independent schrödinger equation be proven? Do you know any source which explains it clearly? Thanks for replies.
You can't prove this equation (you can't prove F=m*a, either), but you can motivate it.
 

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