Prove that for a wave function to be a simultaneous eigenfunction of two operators A and...

Prove that for a wave function to be a simultaneous eigenfunction of two operators A and C, the operators must commute.

Expert Solution

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian or equivalently the Hamiltonian , a function of the generalized coordinates q, generalized velocities and its conjugate momenta:

If either L or H are independent of a generalized coordinate q, meaning the L and H do not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry). Operators in classical mechanics are related to these symmetries.

More technically, when H is invariant under the action of a certain group of transformations G:

.

the elements of G are physical operators, which map physical states among themselves.

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queen_honey_blossom answered 10 months ago

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