Time-Independent Schrödinger Equation

One way is as you suggest: show that the time-evolution operator commutes with the Hamiltonian, which means that the Hamiltonian doesn't change with time. Another way is to observe that if you know the energy of a state, you can figure out its time-evolution.In summary, the time-independent Schrödinger equation is derived from the general Schrödinger equation for time-independent Hamiltonians. It allows for the separation of the time component of the state vector and provides the basis for the vector space of possible physical states and their corresponding energy values. It can be motivated by showing that the time-evolution operator commutes with the Hamiltonian or by observing that knowing the energy of a state allows for determining its
  • #1
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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  • #2
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.
 
  • #3
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.
 

FAQ: Time-Independent Schrödinger Equation

What is the Time-Independent Schrödinger Equation?

The Time-Independent Schrödinger Equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in terms of its wave function. It is a partial differential equation that relates the time-independent Hamiltonian operator to the energy of the system.

What is the significance of the Time-Independent Schrödinger Equation?

The Time-Independent Schrödinger Equation is significant because it allows us to calculate the energy spectrum of a quantum system and predict the probability of finding the system in a particular state. It is also used to study the behavior of atoms, molecules, and other quantum systems.

What are the main assumptions of the Time-Independent Schrödinger Equation?

The Time-Independent Schrödinger Equation assumes that the system is in a stationary state, meaning that its properties do not change with time. It also assumes that the system can be described by a wave function, and that the wave function satisfies the Schrödinger Equation.

How is the Time-Independent Schrödinger Equation solved?

The Time-Independent Schrödinger Equation can be solved using various mathematical techniques, such as separation of variables, perturbation theory, and numerical methods. The solution of the equation gives us the wave function of the system, which contains all the information about its energy and probability distribution.

What are the limitations of the Time-Independent Schrödinger Equation?

The Time-Independent Schrödinger Equation only applies to non-relativistic quantum systems, and it does not take into account the effects of time-varying potentials. It also assumes that the system is in a stationary state, which may not always be the case in real-world scenarios. Additionally, the equation may not accurately describe systems with strong interactions or high energies.

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