The discussion focuses on solving the quantum mechanics problem of a particle in a rigid box when one wall is destroyed at time t=0. The initial wavefunction, ψ(0), is set to zero at the left wall, indicating the particle is in its ground state. To find the evolution of ψ(x, t), the wavefunction must be expanded in terms of free particle plane waves with positive momentum, while considering the boundary conditions imposed by the remaining wall. The absence of translation symmetry means momentum is not a good quantum number, but momentum squared remains valid, allowing for the combination of eigenfunctions of both positive and negative momentum to satisfy the Hamiltonian's boundary conditions.
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Due to the wall, there is no translation symmetry and momentum is not a good quantum number. However, momentum squared is. Combining eigenfunctions of positive and negative momentum you can get eigenfunctions of the Hamiltonian that satisfy the appropriate boundary condition.Christopher Grayce said:
