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complement Alf 1. Distribution of the momentum values in a stationary state a. Cotalation of the function \( \bar{\phi}_{n}(p) \), of \( \langle P\rangle \) and of \( \Delta P \) b. Discristion 2. Evolution of the particle's wave function 3. Wave function at the instant \( t \) b. Ewolution of the shape of the wave packer c. Motion of center of the wave packet 3. Perrurbation created by a position measurement In complement \( \mathrm{H}_{\mathrm{t}}(\S 2-c-\beta \) ), we studied the stationary states of a particle in a onedimensional infinite potential well. Here we intend to re-examine this subject from a physical point of view. This will allow us to apply some of the postulates of chapter III to a concrete case. We shall be particularly interested in the results that can be obtained when the position or momentum of the particle is measured. 1. Distribution of the momentum values in a stationary state 8. CALCULATION OF THE FUNCTION \( \bar{\varphi}_{\odot}(p) \), OF <P> AND OF \( A P \) We have seen that the stationary states of the particle correspond to the energies: \[ E_{n}=\frac{n^{2} \pi^{2} \hbar^{2}}{2 m a^{2}} \] and to the wave functions: \[ \varphi_{n}(x)=\sqrt{\frac{2}{a}} \sin \left(\frac{n \pi x}{a}\right) \] (where \( a \) is the width of the well and \( n \) is any positive integer). Consider a particle in the state \( \left|\varphi_{n}\right\rangle \), with energy \( E_{n} \). The probability of a measurement of the momentum \( P \) of the particle yielding a result between \( p \) and \( p+d p \) is: \[ \bar{S}_{R}(p) \mathrm{d} p=\left|\bar{\varphi}_{n}(p)\right|^{2} \mathrm{~d} p \] with: \[ \bar{\varphi}_{n}(p)=\frac{1}{\sqrt{2 \pi \hbar}} \int_{0}^{a} \sqrt{\frac{2}{a}} \sin \left(\frac{n \pi x}{a}\right) \mathrm{e}^{-i p x / \hbar} \mathrm{d} x \] We shall use the notation of complement \( \mathrm{H}_{\mathrm{r}} \). 269

          complement Alf
1. Distribution of the momentum values in a stationary state
a. Cotalation of the function \( \bar{\phi}_{n}(p) \), of \( \langle P\rangle \) and of \( \Delta P \)
b. Discristion
2. Evolution of the particle's wave function
3. Wave function at the instant \( t \)
b. Ewolution of the shape of the wave packer
c. Motion of center of the wave packet
3. Perrurbation created by a position measurement

In complement \( \mathrm{H}_{\mathrm{t}}(\S 2-c-\beta \) ), we studied the stationary states of a particle in a onedimensional infinite potential well. Here we intend to re-examine this subject from a physical point of view. This will allow us to apply some of the postulates of chapter III to a concrete case. We shall be particularly interested in the results that can be obtained when the position or momentum of the particle is measured.
1. Distribution of the momentum values in a stationary state
8. CALCULATION OF THE FUNCTION \( \bar{\varphi}_{\odot}(p) \), OF <P> AND OF \( A P \)

We have seen that the stationary states of the particle correspond to the energies*:
\[
E_{n}=\frac{n^{2} \pi^{2} \hbar^{2}}{2 m a^{2}}
\]
and to the wave functions:
\[
\varphi_{n}(x)=\sqrt{\frac{2}{a}} \sin \left(\frac{n \pi x}{a}\right)
\]
(where \( a \) is the width of the well and \( n \) is any positive integer).
Consider a particle in the state \( \left|\varphi_{n}\right\rangle \), with energy \( E_{n} \). The probability of a measurement of the momentum \( P \) of the particle yielding a result between \( p \) and \( p+d p \) is:
\[
\bar{S}_{R}(p) \mathrm{d} p=\left|\bar{\varphi}_{n}(p)\right|^{2} \mathrm{~d} p
\]
with:
\[
\bar{\varphi}_{n}(p)=\frac{1}{\sqrt{2 \pi \hbar}} \int_{0}^{a} \sqrt{\frac{2}{a}} \sin \left(\frac{n \pi x}{a}\right) \mathrm{e}^{-i p x / \hbar} \mathrm{d} x
\]
*We shall use the notation of complement \( \mathrm{H}_{\mathrm{r}} \).
269

complement Alf
1. Distribution of the momentum values in a stationary state
a. Cotalation of the function ϕ̅n(p), of ⟨ P⟩ and of Δ P
b. Discristion
2. Evolution of the particle's wave function
3. Wave function at the instant t
b. Ewolution of the shape of the wave packer
c. Motion of center of the wave packet
3. Perrurbation created by a position measurement

In complement Ht( 2-c-β ), we studied the stationary states of a particle in a onedimensional infinite potential well. Here we intend to re-examine this subject from a physical point of view. This will allow us to apply some of the postulates of chapter III to a concrete case. We shall be particularly interested in the results that can be obtained when the position or momentum of the particle is measured.
1. Distribution of the momentum values in a stationary state
8. CALCULATION OF THE FUNCTION φ̅⊙(p), OF <P> AND OF A P

We have seen that the stationary states of the particle correspond to the energies*:

En=(n^2π^2ħ^2)/(2 m a^2)

and to the wave functions:

φn(x)=√((2)/(a))sin((n π x)/(a))

(where a is the width of the well and n is any positive integer).
Consider a particle in the state |φn⟩, with energy En. The probability of a measurement of the momentum P of the particle yielding a result between p and p+d p is:

S̅R(p) d p=|φ̅n(p)|^2 d p

with:

φ̅n(p)=(1)/(√(2 πħ))∫0^a√((2)/(a))sin((n π x)/(a)) e^-i p x / ħd x

*We shall use the notation of complement Hr.
269

Added by David G.

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