A particle of mass, m, moves freely inside an infinite potential well spanning the range, 0
A particle of mass, m, moves freely inside an infinite potential well spanning the range, 0
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![A particle of mass, \( m \), moves freely inside an infinite potential well spanning the range, \( 0 < x < b \). Somehow, we know that the initial wave function for the particle at \( t = 0 \) is given as:
\[
\psi(x, 0) = \sqrt{\frac{2}{7}} \phi_1(x) - \frac{\sqrt{3}}{\sqrt{7}} \phi_2(x) + \frac{\sqrt{9}}{\sqrt{7}} \phi_3(x) - \frac{\sqrt{2}}{\sqrt{7}} \phi_4(x)
\]
**Problems:**
d) At \( t = 0 \), what are the energies that might get measured, and what are the probabilities?
e) At \( t = \frac{1000 m b^2}{h \pi^2} \), what is the average energy?
f) At \( t = \frac{1000 m b^2}{h \pi^2} \), what is the second moment of the energy?
g) Calculate the average energy of the system as a function of time.
h) If the energy of the system is measured to be \( \frac{9 \pi^2 \hbar^2}{2 m b^2} \) at \( t = 0 \), what will \( \psi(x, t) \) equal just after the measurement?
i) If you then measure the energy of the system again at \( t = \frac{1000 m b^2}{h \pi^2} \), what are the energies that might get measured, and what are the probabilities?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e50119e-8646-4255-90fd-98958ba58941%2F225d7ba1-b1b2-4e04-a0d7-6bf7bd78c53d%2Fozepqa5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A particle of mass, \( m \), moves freely inside an infinite potential well spanning the range, \( 0 < x < b \). Somehow, we know that the initial wave function for the particle at \( t = 0 \) is given as:
\[
\psi(x, 0) = \sqrt{\frac{2}{7}} \phi_1(x) - \frac{\sqrt{3}}{\sqrt{7}} \phi_2(x) + \frac{\sqrt{9}}{\sqrt{7}} \phi_3(x) - \frac{\sqrt{2}}{\sqrt{7}} \phi_4(x)
\]
**Problems:**
d) At \( t = 0 \), what are the energies that might get measured, and what are the probabilities?
e) At \( t = \frac{1000 m b^2}{h \pi^2} \), what is the average energy?
f) At \( t = \frac{1000 m b^2}{h \pi^2} \), what is the second moment of the energy?
g) Calculate the average energy of the system as a function of time.
h) If the energy of the system is measured to be \( \frac{9 \pi^2 \hbar^2}{2 m b^2} \) at \( t = 0 \), what will \( \psi(x, t) \) equal just after the measurement?
i) If you then measure the energy of the system again at \( t = \frac{1000 m b^2}{h \pi^2} \), what are the energies that might get measured, and what are the probabilities?
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