Representing a Hamiltonian in an operator form

[nashed]

Given a Hamiltonian in the position representation how do I represent it in operator form? for example I was asked to calculate the expectancy of the Darwin correction to the Hydrogen Hamiltonian given some eigenstate (I think it was |2,1> or something bu that doesn't matter right now), now I know that the Hamiltonian is given in the position basis and as such I calculated the relevant integral, but the thing is that I wanted to try and do it algebraically and didn't know how to proceed...

 

[dextercioby]

The Hamiltonian is always given in an operator form, i.e. it is a (rigged)-Hilbert space operator written as a function (typically algebraic) of other operators, such as position, momentum, angular momentum (including spin), electric charge, parity, etc. Only the Hilbert space is mapped to a certain space (a function space) so that the Hamiltonian and all other observables could be represented by differential operators. The so-called algebraic methods work by assuming no "projection onto a function space" takes place. This works for any Hamiltonian with (a) discrete (part of a) spectrum. This is easiest to see for the harmonic oscillator, but it also works for the discrete spectrum of the H-atom Hamiltonian.

1st step: identify the exact form of the Darwin correction term. IIRC, this is expressible only in a Hilbert space of wave functions, i.e. the "projection onto a function space" $|\psi\rangle \rightarrow \langle x|\psi\rangle$ already took place.
2nd step: project the abstract vector $|2,1\rangle$ onto the same basis as the Hamiltonian was projected.
3rd step: compute the matrix elements.

 

[nashed]

The Hamiltonian is always given in an operator form, i.e. it is a (rigged)-Hilbert space operator written as a function (typically algebraic) of other operators, such as position, momentum, angular momentum (including spin), electric charge, parity, etc. Only the Hilbert space is mapped to a certain space (a function space) so that the Hamiltonian and all other observables could be represented by differential operators. The so-called algebraic methods work by assuming no "projection onto a function space" takes place. This works for any Hamiltonian with (a) discrete (part of a) spectrum. This is easiest to see for the harmonic oscillator, but it also works for the discrete spectrum of the H-atom Hamiltonian.

1st step: identify the exact form of the Darwin correction term. IIRC, this is expressible only in a Hilbert space of wave functions, i.e. the "projection onto a function space" $|\psi\rangle \rightarrow \langle x|\psi\rangle$ already took place.
2nd step: project the abstract vector $|2,1\rangle$ onto the same basis as the Hamiltonian was projected.
3rd step: compute the matrix elements.

Thanks for the help, but in the case of the Darwin term it's given by some constant times a delta function, how am I supposed to get the operator form of this expression? another example is the spin-orbital interaction, it's given by 1/r^3 times some constant time the angular momentum operator dotted with the spin operator, how am I supposed to treat the 1/r^3 part? as a scalar or as an operator, also if it's an operator how do I treat as it is not an analytic function of R

 

[DrClaude]

another example is the spin-orbital interaction, it's given by 1/r^3 times some constant time the angular momentum operator dotted with the spin operator, how am I supposed to treat the 1/r^3 part?

Some times, operators only have a simple form in a particular representation, and one has to use that representation to get the effect of the operator on a particular ket.

 

[blue_leaf77]

Thanks for the help, but in the case of the Darwin term it's given by some constant times a delta function, how am I supposed to get the operator form of this expression?

Is it not $\delta(\hat{\mathbf r})$?

Thanks for the help, but in the case of the Darwin term it's given by some constant times a delta function, how am I supposed to get the operator form of this expression? another example is the spin-orbital interaction, it's given by 1/r^3 times some constant time the angular momentum operator dotted with the spin operator, how am I supposed to treat the 1/r^3 part? as a scalar or as an operator, also if it's an operator how do I treat as it is not an analytic function of R

There is a good discussion on how these various correction terms were derived in appendix of the book Physics of Atoms and Molecules by B. H. Bransden and C. J. Joachain. The fact that they are correction terms means that some degree of exactness have been reduced out of the original exact, operator forms. Therefore, they do not necessarily have the corresponding operator form. The exact treatment of relativistic effects in hydrogen-like atoms can be found in the 2nd edition of Modern Quantum Mechanics by Sakurai and Napolitano.