cemtu
- 99
- 7
- Homework Statement
- How to show that eigenvalues of 3p state of a hydrogen atom are perpendicular to each other?
- Relevant Equations
- I am not sure if this solution is correct
This is a general property of eigenvectors of Hermitian operators. State functions are a particular class of vector, and it is easiest to work in the general formalism (I am hoping to show how ket notation makes qm easier, not just do standard bookwork at this level). Suppose O is a Hermitian operator with eigenvalues k and l with k≠l corresponding to eigenstates |k⟩ and |l⟩
O|k⟩=k|k⟩(1)
O|l⟩=l|l⟩(2)
For the present case, we are interested in the orthogonality of angular momentum states, so O can be taken to be Lz, but the principle is completely general.
Act on https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq1 with ⟨k|
⟨k|O|k⟩=k⟨k|k⟩
It follows at once that k is real (as are all eigenvalues of a Hermitian operator). Act on https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq2 with ⟨k|
⟨k|O|l⟩=l⟨k|l⟩
Take the complex conjugate
⟨l|O|k⟩=l⟨l|k⟩(3)
Act on https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq1 with ⟨l|⟨l|O|k⟩=k⟨l|k⟩(4)
Subtract https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq3 from https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq4
(k−l)⟨l|k⟩=0
Since k≠l, |k⟩ and |l⟩ are orthogonal (not perpendicular, that applies to ordinary 3-vectors).
O|k⟩=k|k⟩(1)
O|l⟩=l|l⟩(2)
For the present case, we are interested in the orthogonality of angular momentum states, so O can be taken to be Lz, but the principle is completely general.
Act on https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq1 with ⟨k|
⟨k|O|k⟩=k⟨k|k⟩
It follows at once that k is real (as are all eigenvalues of a Hermitian operator). Act on https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq2 with ⟨k|
⟨k|O|l⟩=l⟨k|l⟩
Take the complex conjugate
⟨l|O|k⟩=l⟨l|k⟩(3)
Act on https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq1 with ⟨l|⟨l|O|k⟩=k⟨l|k⟩(4)
Subtract https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq3 from https://physics.stackexchange.com/questions/547478/how-to-show-that-eigenfunctions-of-3p-state-of-a-hydrogen-atom-are-perpendicular#mjx-eqn-eq4
(k−l)⟨l|k⟩=0
Since k≠l, |k⟩ and |l⟩ are orthogonal (not perpendicular, that applies to ordinary 3-vectors).
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