Conservation laws in Schrodinger equations

physics.stackexchange.com/questions/229885/energy-equation-in-quantum-mechanics

TODO is there any good intuitive argument or proof of conservation of energy, momentum, angular momentum?

Table of contents
- Conservation of the square amplitude in the Schrodinger equation Conservation laws in Schrodinger equations
  - Probability current Conservation of the square amplitude in the Schrodinger equation

Conservation of the square amplitude in the Schrodinger equation

Proof that the probability 1 is conserved by the time evolution:

It can be derived directly from the Schrödinger equation.

Bibliography:

quantummechanics.ucsd.edu/ph130a/130_notes/node127.html from quantum physics by Jim Branson (2003).
That proof also mentions that if the potential V is not real, then there is no conservation of probability! Therefore the potential must be real valued!

Probability current

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