Potential operator in positional space

[qubits]

Is the potential operator (in positional/space basis) of the Hamiltonian always diagonal in that basis? And is the kinetic energy operator always diagonal in complementary momentum space?

 

[Eye_in_the_Sky]

For a spinless particle in one dimension, the most general Hamiltonian which satisfies Galilean invariance is

H_t = P²/2m + V_t(Q) .

So, in this case, the answer to your question is "yes".
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However, for the case of a spinless particle in three dimensions, the most general Hamiltonian which satisfies Galilean invariance is

H_t = |P - A_t(Q)|²/2m + V_t(Q) .

The first term on the right hand side is the "kinetic" term ... clearly, it is not diagonal in the momentum representation.

 

[R0man]

The potential operator in positional space refers to the mathematical representation of the potential energy in terms of the position of a particle. It is an operator that acts on the wavefunction of a system to determine the potential energy at a given position.

The answer to whether the potential operator of the Hamiltonian is always diagonal in positional space is dependent on the specific system being studied. In some cases, the potential energy may only depend on the position of the particle, making the potential operator diagonal. However, in other cases, the potential energy may depend on other variables such as time or other parameters, making the potential operator non-diagonal.

Similarly, the kinetic energy operator in complementary momentum space may or may not be diagonal. In some systems, the kinetic energy may only depend on the momentum of the particle, making the kinetic energy operator diagonal. However, in other systems, the kinetic energy may depend on other variables such as position or spin, making the kinetic energy operator non-diagonal.

In summary, the diagonal nature of the potential and kinetic energy operators in positional and momentum space is dependent on the specific system being studied. In general, these operators can be either diagonal or non-diagonal, and it is important to consider the specific properties of the system when determining their nature.