Advanced Physcs 10
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School
Purdue University *
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Course
272
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
2
Uploaded by DoctorRabbitMaster510
**1. Question:**
A particle is described by the wave function \( \Psi(x, t) = Ae^{i(kx - \omega t)} \), where \( A \) is
a constant, \( k \) is the wave number, and \( \omega \) is the angular frequency. Determine the
probability density \( |\Psi(x, t)|^2 \) and find the average position \( \langle x \rangle \) of the
particle.
**Answer:**
The probability density is given by \( |\Psi(x, t)|^2 = \Psi^*(x, t) \Psi(x, t) = |A|^2 \). Thus, the
probability density is independent of position and time, indicating a constant probability
distribution. The average position \( \langle x \rangle \) is found by integrating the probability
density over all space:
\[ \langle x \rangle = \int_{-\infty}^{\infty} x |\Psi(x, t)|^2 dx = \int_{-\infty}^{\infty} x |A|^2 dx = |A|^2
\int_{-\infty}^{\infty} x dx \]
\[ = |A|^2 \left[ \frac{x^2}{2} \right]_{-\infty}^{\infty} = |A|^2 \left( \frac{\infty - (-\infty)}{2} \right) =
\infty \]
Thus, the average position of the particle is infinite, indicating that it is not confined to any
particular region of space.
**2. Question:**
A block of mass \( m \) is attached to a spring with spring constant \( k \) and is oscillating in
simple harmonic motion with amplitude \( A \). Determine the maximum speed \( v_{\text{max}}
\) of the block in terms of \( A \) and the angular frequency \( \omega \).
**Answer:**
The maximum speed occurs when the displacement \( x \) from equilibrium is equal to the
amplitude \( A \). At this point, the block experiences maximum acceleration, which corresponds
to the maximum speed. Using the equation for simple harmonic motion \( x(t) = A \cos(\omega t)
\), we can find the maximum speed:
\[ v_{\text{max}} = \frac{d}{dt} x(t) \bigg|_{x=A} = -A \omega \sin(\omega t) \bigg|_{x=A} = A
\omega \]
Thus, the maximum speed \( v_{\text{max}} \) of the block is \( A \omega \).
**3. Question:**
A particle of mass \( m \) is confined to a one-dimensional box of length \( L \). Determine the
allowed energy levels \( E_n \) of the particle.
**Answer:**
In a one-dimensional box, the energy levels are quantized and given by the equation:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \]
where \( n \) is a positive integer representing the quantum number, \( \hbar \) is the reduced
Planck's constant, and \( m \) is the mass of the particle. The energy levels \( E_n \) increase as
the quantum number \( n \) increases, with the lowest energy level \( E_1 \) corresponding to \( n
= 1 \).
**4. Question:**
A charged particle of mass \( m \) and charge \( q \) moves in a uniform magnetic field \(
\mathbf{B} \) with velocity \( \mathbf{v} \). Derive the equation of motion for the particle.
**Answer:**
The Lorentz force acting on the charged particle is given by \( \mathbf{F} = q (\mathbf{v} \times
\mathbf{B}) \), where \( \times \) denotes the cross product. Using Newton's second law \(
\mathbf{F} = m \mathbf{a} \), where \( \mathbf{a} \) is the acceleration of the particle, we can
write:
\[ m \mathbf{a} = q (\mathbf{v} \times \mathbf{B}) \]
\[ \mathbf{a} = \frac{q}{m} (\mathbf{v} \times \mathbf{B}) \]
This equation describes the motion of the charged particle in a uniform magnetic field.
**5. Question:**
A particle is described by the wave function \( \Psi(x, t) = A e^{-\frac{x^2}{2\sigma^2}}
e^{-i\omega t} \), where \( A \) is a constant, \( \sigma \) is the standard deviation, and \( \omega
\) is the angular frequency. Determine the normalization constant \( A \) and find the uncertainty
in position \( \Delta x \).
**Answer:**
To find the normalization constant \( A \), we must ensure that the integral of the probability
density \( |\Psi(x, t)|^2 \) over all space is equal to 1. This gives:
\[ \int_{-\infty}^{\infty} |\Psi(x, t)|^2 dx = |A|^2 \int_{-\infty}^{\infty} e^{-\frac{x^2}{\sigma^2}} dx =
|A|^2 \sqrt{\pi} \sigma = 1 \]
\[ \Rightarrow A = \left( \frac{1}{\sqrt{\sqrt{\pi} \sigma}} \right)^{\frac{1}{2}} \]
The uncertainty in position \( \Delta x \) is given by twice the standard deviation \( \sigma \), i.e.,
\( \Delta x = 2 \sigma \).
**6. Question:**
A particle of mass \( m \) is in a one-dimensional potential well described by the potential
function \( V(x) = V_0 \sin^2(\frac{\pi x}{a}) \), where \( V_0 \) is the amplitude of the potential
and \( a \) is the width of the well. Determine the allowed energy levels \( E_n \) of the particle.
**Answer:**
The allowed energy levels of the particle in the one-dimensional potential well are quantized and
given by the equation:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2 m a^2} \]
where \( n \) is a positive integer representing the quantum number. These energy levels
correspond to the discrete eigenvalues of the Schrödinger equation for the particle in the
potential well. The lowest energy level \( E_1 \) corresponds to \( n = 1 \), and the energy levels
increase with increasing values of \( n \).
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