- #1
Raz91
- 21
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Prove the equation
[itex]A\left|\psi\right\rangle = \left\langle A\right\rangle\left|\psi\right\rangle + \Delta A\left|\psi\bot\right\rangle[/itex]
where [itex] A[/itex] is a Hermitian operator and [itex] \left\langle\psi |\psi\bot\right\rangle = 0[/itex]
[itex]\left\langle A\right\rangle[/itex] = The expectation value of A.
[itex]\Delta A[/itex] = The standard deviation of A.
My attempt :
I tried to write [itex]\left|\psi\right\rangle[/itex] as a superposition of the eigenfuncion of the operator [itex] A [/itex] and used the fact that it's a Hermitian operator
[itex]A\left|\phi_{n}\right\rangle = \lambda_{n}\left|\phi_{n}\right\rangle[/itex] , [itex]\left|\psi\right\rangle = \sum a_{n}\left|\phi_{n}\right\rangle[/itex]
so that [itex]A\left|\psi\right\rangle = \sum a_{n}\lambda_{n}\left|\phi_{n}\right\rangle[/itex]
and [itex]\left\langle A\right\rangle = \sum |a_{n}|^{2}\lambda_{n} [/itex]
[itex]\Delta A = \sqrt{\left\langle A^{2}\right\rangle - \left\langle A\right\rangle ^{2}}[/itex]
and I wrote [itex]\left|\psi\bot\right\rangle[/itex] as [itex]\left|\psi\bot\right\rangle = \sum b_{n}\left|\phi_{n}\right\rangle[/itex]
[itex]\sum a^{*}_{n}b_{n} = 0[/itex]
I don't know how to go on from here...
any ideas?
thank you! :)
[itex]A\left|\psi\right\rangle = \left\langle A\right\rangle\left|\psi\right\rangle + \Delta A\left|\psi\bot\right\rangle[/itex]
where [itex] A[/itex] is a Hermitian operator and [itex] \left\langle\psi |\psi\bot\right\rangle = 0[/itex]
[itex]\left\langle A\right\rangle[/itex] = The expectation value of A.
[itex]\Delta A[/itex] = The standard deviation of A.
My attempt :
I tried to write [itex]\left|\psi\right\rangle[/itex] as a superposition of the eigenfuncion of the operator [itex] A [/itex] and used the fact that it's a Hermitian operator
[itex]A\left|\phi_{n}\right\rangle = \lambda_{n}\left|\phi_{n}\right\rangle[/itex] , [itex]\left|\psi\right\rangle = \sum a_{n}\left|\phi_{n}\right\rangle[/itex]
so that [itex]A\left|\psi\right\rangle = \sum a_{n}\lambda_{n}\left|\phi_{n}\right\rangle[/itex]
and [itex]\left\langle A\right\rangle = \sum |a_{n}|^{2}\lambda_{n} [/itex]
[itex]\Delta A = \sqrt{\left\langle A^{2}\right\rangle - \left\langle A\right\rangle ^{2}}[/itex]
and I wrote [itex]\left|\psi\bot\right\rangle[/itex] as [itex]\left|\psi\bot\right\rangle = \sum b_{n}\left|\phi_{n}\right\rangle[/itex]
[itex]\sum a^{*}_{n}b_{n} = 0[/itex]
I don't know how to go on from here...
any ideas?
thank you! :)