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## Main Question or Discussion Point

i can't seem to understand something very basic about the stationary equation (a "simple" eigenvalue problem):

H|Y>=E|Y>

H - hamiltonian operator

Y - an eigenstate or an eigenfunction of the hamiltonian

E - the eigenvalue of the eigenstate

as far as i understand, the hamilotian operator (H) is represented in the energy basis, and the eigenstates (Y) therefore form a basis that span the hilbert space of the system.

now this is where i get fairly confused. if the equation above is multiplied from the left by <x|, a bra representing one of the position states that form a position basis, then the equation is written as

H*Y(x)=E*Y(x)

BUT, it is also said that the hamiltonian is represented in the energy basis, so why is "x" being the variable in use here?

it seems as if the operator is in one basis, but the state is in a different one, but i know that i am wrong i just can't figure this out

i hope someone can help me here...

H|Y>=E|Y>

H - hamiltonian operator

Y - an eigenstate or an eigenfunction of the hamiltonian

E - the eigenvalue of the eigenstate

as far as i understand, the hamilotian operator (H) is represented in the energy basis, and the eigenstates (Y) therefore form a basis that span the hilbert space of the system.

now this is where i get fairly confused. if the equation above is multiplied from the left by <x|, a bra representing one of the position states that form a position basis, then the equation is written as

H*Y(x)=E*Y(x)

BUT, it is also said that the hamiltonian is represented in the energy basis, so why is "x" being the variable in use here?

it seems as if the operator is in one basis, but the state is in a different one, but i know that i am wrong i just can't figure this out

i hope someone can help me here...