**Problem 48:**From equations (6-23) and (6-29) obtain the dispersion coefficient for matter waves (in vacuum), then show that probability density (6-35) follows from (6-28). ### Transcription and Explanation for an Educational Website#### Equations and Definitions from Quantum Mechanics**Equation (6-35):**The probability density is given by:\[|\Psi(x, t)|^2 = \frac{C^2}{\sqrt{1 + \hbar^2 t^2 / 4m^2 \varepsilon^4}} \exp \left[ -\frac{(x - st)^2}{2 \varepsilon^2 (1 + \hbar^2 t^2 / 4m^2 \varepsilon^4)} \right]\]**Equation (6-28):**Another form of the probability density is expressed as:\[|\Psi(x, t)|^2 = \frac{C^2}{\sqrt{1 + D^2 t^2 / 4e^4}} \exp \left[ -\frac{(x - st)^2}{2e^2 (1 + D^2 t^2 / 4e^4)} \right]\]**Definitions (from Equations 6-29 and 6-23):**- \( s = \left. \frac{d\omega(k)}{dk} \right|_{k_0} \)- \( D = \left. \frac{d^2 \omega(k)}{dk^2} \right|_{k_0} \)- Matter wave dispersion relation: \[ \omega(k) = \frac{\hbar k^2}{2m} \]### ExplanationThese equations represent the probability density of a quantum particle's position over time, described by a wave packet. Such representations are crucial in understanding wave mechanics within quantum theory.- **\( \Psi(x, t) \)**: This function represents the wave function of a particle at position \( x \) and time \( t \).- **\( C \)**: A constant that is related to the normalization of the wave function.- **\( s \) and \( D \)**: Parameters that derive from the dispersion relation of the wave, which determine the group and phase velocity.- **\( \varepsilon \) and \( e \)**: Parameters related to the spread or width of the wave packet.- **\( \hbar \)**: Reduced Planck's constant.- **\( m \)**: Mass of

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Answered: 48. From equations (6-23) and (6-29)…

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48. From equations (6-23) and (6-29) obtain the dispersion coefficient for matter waves (in vacuum), then show that probability density (6-35) follows from (6-28).