Like in the Hamiltonian for a particle in an electromagnetic field. This is not a conservative field so the Hamiltonian doesn't represent the energy of the system. And yet the time independent Schrodinger equation still reads $H \psi = E \psi$ (for example that's what Griffiths did in page 183 of his book). Which suggests that the Hamiltonian does represent the energy of the system. What am I missing?
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You are quite right to say that the Hamiltonian is not always the energy in the sense of being kinetic plus potential energy. This statement is often repeated in courses on analytical mechanics, to drill the idea that the Hamiltonian is the energy function in specific variables, namely position and momentum. But it is always a conserved quantity, and the actual values the Hamiltonian takes are energy values. Since eigenvalues are the values of the Hamiltonian that can be measured, and since these values do not depend on the variables in which they were expressed, they can be called energies.
On the other hand, your saying that the electromagnetic field is not conservative is possibly a misunderstanding: For velocity-independent forces we call the force field conservative if it does no work on a particle moving on a closed circuit. This definition fails for the Lorentz-force, which I assume you have in mind, because it never performs any work on a moving particle.
As an example, for a particle in a magnetic field, the energy is always $$ \frac m2(v_x^2+v_y^2) $$
But the Hamiltonian is expressed in terms of a nonstandard momentum, and is given by $$ \frac1{2m}\left[(p_x - \frac{e B}{m c}y)^2 + p_y^2\right] $$
but it has the same values as the energy. Its expression is different because we deal with momenta.
leyvraz
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A non-conservative field just means that energy can be added or removed from the system. It does not change the fact that the Hamiltonian represents energy.
Charles Francis
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