Quantum mechanics MOC

QM in 3D position-space

Consider the Hilbert space with vectors represented in the position basis . The momentum operators are given by

and thus the Hamiltonian operator by

and the Schrödinger equation becomes

Time independent Schrödinger equation

If is time-independent the stationary states are given by the time-independent Schrödinger equation

and thus general solutions are given by

Properties

  1. The canonical commutation relations are

an example of the Standard Heisenberg algebra for QM. 2. 3. The energy of a normalizable solution must exceed the infimum of the potential.

Proof of 1–2

For any with

as required

Since any normalizable solution is a linear combination of stationary states, it is sufficient to show all stationary states have definite energy greater than this infimum. According to the Time independent Schrödinger equation

If for all then never has the opposite sign to . If is positive then is concave up, and if is negative then is concave down. Hence never approaches zero as .

Spherical coördinates

In Spherical coördinates the Hamiltonian is

Examples

See also

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