- #1
raintrek
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I'm working through a proof of the orthormality condition for a complete set of states and am struggling with one element of it:
Consider the eigenstates of the Hamiltonian in the following way:
1: [tex]\int\Psi^{*}_{m}H\Psi_{n}dV = E_{n}\int\Psi^{*}_{m}H\Psi_{n}dV[/tex]
and
2: [tex]\int\Psi^{*}_{n}H\Psi_{m}dV = E_{m}\int\Psi^{*}_{n}H\Psi_{m}dV[/tex]
Taking the complex conjugate of the second (for H real) to obtain
3: [tex]\int\Psi_{n}H\Psi^{*}_{m}dV = E_{m}\int\Psi_{n}H\Psi^{*}_{m}dV[/tex]
Now subtract them:
1-3: [tex]\int\Psi^{*}_{m}H\Psi_{n}dV - \int\Psi^{*}_{n}H\Psi_{m}dV = (E_{n} - E_{m}) \int\Psi^{*}_{m}H\Psi_{n}dV[/tex]
I don't understand why the LHS of eq. 3 in this subtraction has had it's complex conjugate taken again, yet the right hand side has remained as it is in 3. Is it something to do with the Hermiticity of the Hamiltonian? Many thanks
Consider the eigenstates of the Hamiltonian in the following way:
1: [tex]\int\Psi^{*}_{m}H\Psi_{n}dV = E_{n}\int\Psi^{*}_{m}H\Psi_{n}dV[/tex]
and
2: [tex]\int\Psi^{*}_{n}H\Psi_{m}dV = E_{m}\int\Psi^{*}_{n}H\Psi_{m}dV[/tex]
Taking the complex conjugate of the second (for H real) to obtain
3: [tex]\int\Psi_{n}H\Psi^{*}_{m}dV = E_{m}\int\Psi_{n}H\Psi^{*}_{m}dV[/tex]
Now subtract them:
1-3: [tex]\int\Psi^{*}_{m}H\Psi_{n}dV - \int\Psi^{*}_{n}H\Psi_{m}dV = (E_{n} - E_{m}) \int\Psi^{*}_{m}H\Psi_{n}dV[/tex]
I don't understand why the LHS of eq. 3 in this subtraction has had it's complex conjugate taken again, yet the right hand side has remained as it is in 3. Is it something to do with the Hermiticity of the Hamiltonian? Many thanks