Representing Operators as Matrices and Differential Operators

[aaaa202]

An operator A defined by a matrix can be written as something like:

A = Ʃ_i,jle_i><e_jl <e_ilAle_j>

How does this representation translate to a continuous basis, e.g. position basis, where operators are not matrices but rather differential operators etc. Can we still write for e.g. the kinetic energy operator T:

T = ∫∫dr dr' lr'><rl <r'lTlr>

? Or how would T be represented.

 

[Chopin]

I know there are some subtleties to shifting to a continuous basis, but basically yes. You change $\Sigma$'s into $\int$'s, and then all the same logic goes through. $T$ becomes a "matrix" with an infinite number of rows and columns, which you could think of as a function $T(q_1, q_2)$.

 

[aaaa202]

The energy eigenstates form a countable basis. Would it be possible to do the above expansion in those rather than position eigenstates?

 

[hilbert2]

^ The eigenstates of the Hamiltonian form a countable basis only if all the states are bound states, like in a harmonic oscillator. For example, the bound states of hydrogen atom form a discrete set, but the system also has a continuous spectrum of scattering states which have positive total energy. The continuous spectrum is not a countable set.

 

[aaaa202]

but either way you can expand any operator in energy just as well as position eigenstates right? or any other observables eigenstates..

 

[hilbert2]

^ Yes, the eigenstates of any hermitian operator form a complete basis.