- #1

Geremia

- 151

- 0

Let [itex]|\psi_n\rangle\in\mathcal{H}[/itex], where [itex]\mathcal{H}[/itex] is Hilbert space, be orthonormal states forming a complete set, and [itex]n\in\mathbb{N}[/itex]. Let

[tex]|\Psi\rangle=\sum_{n=1}^N c^{(1)}_n|\psi_n\rangle,[/tex]

where [itex]c_n[/itex]s are normalized coefficients and [itex]N[/itex] is either finite or infinite. Let [itex]m[/itex] be an eigenvalue of observable [tex]\hat O[/tex] corresponding to the eigenket [itex]|\phi\rangle\in\mathcal{H}[/itex], where

[tex]|\phi\rangle=\sum_{n=1}^N c^{(2)}_n|\psi_n\rangle.[/tex]

The probability of measuring [itex]m[/itex] is

[tex]|\langle\phi|\Psi\rangle|^2=\frac{\mathrm{occurrence(s)\;of\;measuring\;particular\;state(s)\;}|\psi_n\rangle}{\mathrm{total\;possible\;measurable\;states\;}N},[/tex]

which follows from the definition of probability. What specifically is "occurrence(s) of measuring particular state(s)" for given [itex]c_n[/itex]s and [itex]N[/itex]?

[tex]|\Psi\rangle=\sum_{n=1}^N c^{(1)}_n|\psi_n\rangle,[/tex]

where [itex]c_n[/itex]s are normalized coefficients and [itex]N[/itex] is either finite or infinite. Let [itex]m[/itex] be an eigenvalue of observable [tex]\hat O[/tex] corresponding to the eigenket [itex]|\phi\rangle\in\mathcal{H}[/itex], where

[tex]|\phi\rangle=\sum_{n=1}^N c^{(2)}_n|\psi_n\rangle.[/tex]

The probability of measuring [itex]m[/itex] is

[tex]|\langle\phi|\Psi\rangle|^2=\frac{\mathrm{occurrence(s)\;of\;measuring\;particular\;state(s)\;}|\psi_n\rangle}{\mathrm{total\;possible\;measurable\;states\;}N},[/tex]

which follows from the definition of probability. What specifically is "occurrence(s) of measuring particular state(s)" for given [itex]c_n[/itex]s and [itex]N[/itex]?

Last edited: