- #1

EE18

- 112

- 13

Ballentine, in his Chapter 8.1, appears to give the attached recipe for *in principle* preparing an (almost) arbitrary (pure) state (of a particle with no internal degrees of freedom) by the method of "waiting for decay to the energy ground state". My questions are fourfold:

1) From (8.1), we are clearly constructing the potential ##W_1## so that ##R(\mathbf{x})## is an eigenstate of the corresponding Hamiltonian. However, why on Earth should we expect it to be the ground state energy? Do we need to fix ##E## in a certain way so that this is true? Is there some theorem which guarantees that, if I pick ##E## a certain way, then that will be the lowest eigenstate for the corresponding potential?2) Following this question we are led naturally to ask what restrictions, if any, must be imposed on the assumed ground state energy ##E##, in order that the potentials be physically reasonable? I can't seem to think of any since ##E## is not in a denominator or something like that.

3) Ballentine writes "We must restrict ##R(\mathbf{x})## to be a nodeless function, otherwise it will not be the ground state." Why is this true? I can see from (8.1) that nodes in ##R## may cause trouble (although we seem to divide by functions which take on the value 0 all the time when using the method of separation of variables, so perhaps this isn't the issue). Is there a theorem which says ground states can't have nodes? And how, if at all, does this tie into question 1 (i.e. if it doesn't have a node, how can I guarantee this will be the ground state)?

1) From (8.1), we are clearly constructing the potential ##W_1## so that ##R(\mathbf{x})## is an eigenstate of the corresponding Hamiltonian. However, why on Earth should we expect it to be the ground state energy? Do we need to fix ##E## in a certain way so that this is true? Is there some theorem which guarantees that, if I pick ##E## a certain way, then that will be the lowest eigenstate for the corresponding potential?2) Following this question we are led naturally to ask what restrictions, if any, must be imposed on the assumed ground state energy ##E##, in order that the potentials be physically reasonable? I can't seem to think of any since ##E## is not in a denominator or something like that.

3) Ballentine writes "We must restrict ##R(\mathbf{x})## to be a nodeless function, otherwise it will not be the ground state." Why is this true? I can see from (8.1) that nodes in ##R## may cause trouble (although we seem to divide by functions which take on the value 0 all the time when using the method of separation of variables, so perhaps this isn't the issue). Is there a theorem which says ground states can't have nodes? And how, if at all, does this tie into question 1 (i.e. if it doesn't have a node, how can I guarantee this will be the ground state)?