![]() (5.3) We will now try to express this equation as the square of some (yet unknown) operator p 2 +x 2 → (x + ip)(x − ip) = p 2 +x 2 + i(px −xp), (5.4) but since x and p do not commute (remember Theorem 2.3), we only will succeed by taking the x − p commutator into account. (5.2) by defining the new operatorx := mωx H ψ = 1 2m i d dx 2 + (mωx) 2 ψ = 1 2m p 2 +x 2 ψ = E ψ. 5.1.1 Algebraic Method We start again by using the time independent Schrödinger equation, into which we insert the Hamiltonian containing the harmonic oscillator potential (5.1) H ψ = − 2 2m d 2 dx 2 + mω 2 2 x 2 ψ = E ψ. ![]() for example, in the references below, the time dependent Schrodinger equation is the 5th postulate. ![]() It is not proper to say that it is derived, unless you have a different set of postulates. (5.1) There are two possible ways to solve the corresponding time independent Schrödinger equation, the algebraic method, which will lead us to new important concepts, and the analytic method, which is the straightforward solving of a differential equation. The time dependent Schrodinger equation is one of 5 (or 6) postulates of quantum mechanics. Deriving time independent Schrdinger equation. In this chapter we will study the features of one of the most important potentials in physics, it's the harmonic oscillator potential which is included now in the Hamiltonian V (x) = mω 2 2 x 2. Can be interpreted as the average value of x that we expect to.
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