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8.3 The measurement process

Suppose I know my wave function at time t = 0 is the sum of the two lowest-energy harmonic oscillator wave functions,

$\displaystyle \psi$(x, 0) = $\displaystyle {\frac{1}{\sqrt 2}}$$\displaystyle \left[\vphantom{ \phi_{0}(x)+\phi_{1}(x) }\right.$$\displaystyle \phi_{0}^{}$(x) + $\displaystyle \phi_{1}^{}$(x)$\displaystyle \left.\vphantom{ \phi_{0}(x)+\phi_{1}(x) }\right]$. (8.25)

The introduction of the time independent wave function was through the separation $ \psi_{n}^{}$(x, t) = e-iEn/$\scriptstyle \hbar$t$ \phi_{n}^{}$(x). Together with the superposition for time-dependent wave functions, we find

$\displaystyle \psi$(x, t) = $\displaystyle {\frac{1}{\sqrt 2}}$$\displaystyle \left[\vphantom{ \phi_{0}(x) e^{-i{\textstyle{\frac{1}{2}}}\omega t}+\phi_{1}(x) e^{-i{\textstyle{\frac{3}{2}}}\omega t}}\right.$$\displaystyle \phi_{0}^{}$(x)e-i$\scriptstyle {\frac{1}{2}}$$\scriptstyle \omega$t + $\displaystyle \phi_{1}^{}$(x)e-i$\scriptstyle {\frac{3}{2}}$$\scriptstyle \omega$t$\displaystyle \left.\vphantom{ \phi_{0}(x) e^{-i{\textstyle{\frac{1}{2}}}\omega t}+\phi_{1}(x) e^{-i{\textstyle{\frac{3}{2}}}\omega t}}\right]$. (8.26)

The expectation value of $ \hat{H}$, i.e., the expectation value of the energy is

$\displaystyle \left\langle\vphantom{ \hat H }\right.$$\displaystyle \hat{H}$ $\displaystyle \left.\vphantom{ \hat H }\right\rangle$ = $ {\frac{1}{2}}$(E0 + E1) = $\displaystyle \hbar$$\displaystyle \omega$. (8.27)

The interpretation of probilities now gets more complicated. If we measure the energy, we don't expect an outcome E3, since there is no $ \phi_{3}^{}$ component in the wave functon. We do expect E0 = $ {\frac{1}{2}}$$ \hbar$$ \omega$ or E1 = $ {\frac{3}{2}}$$ \hbar$$ \omega$ with 50 % propability, which leads to the right average. Actually simple mathematics shows that the result for the expectation value was just that, $ \langle$E$ \rangle$ = $ {\frac{1}{2}}$E0 + $ {\frac{1}{2}}$E1.

We can generalise this result to stating that if

$\displaystyle \psi$(x, t) = $\displaystyle \sum_{n=0}^{\infty}$cn(t)$\displaystyle \phi_{n}^{}$(x), (8.28)

where $ \phi_{n}^{}$(x) are the eigenfunctions of an (Hermitean) operator $ \hat{O}$,

$\displaystyle \hat{O}$$\displaystyle \phi_{n}^{}$(x) = on$\displaystyle \phi_{n}^{}$(x), (8.29)

then

$\displaystyle \left\langle\vphantom{ \hat O }\right.$$\displaystyle \hat{O}$ $\displaystyle \left.\vphantom{ \hat O }\right\rangle$ = $\displaystyle \sum_{n=0}^{\infty}$| cn(t)|2, (8.30)

and the probability that the outcome of a measurement of $ \hat{O}$ at time t0 is on is | cn(t)|2. Here we use orthogonality and completeness of the eigenfunctions of Hermitean operators.



 
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© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk