Exercise 7.10 Express the internal energy in terms of reduced distribution functions when \begin{equation*} U(r^N) = \sum_{i>j=1} u(r_{ij}) + \sum_{i>j>l=1} u^{(3)}(r_i - r_j, r_i - r_l). \end{equation*}
Added by Alexander P.
Close
Step 1
The pair distribution function, g(r), is defined as the probability density of finding a pair of particles at a distance r from each other. It is related to the reduced distribution function, u(r), by the equation: g(r) = rho * u(r) where rho is the number Show more…
Show all steps
Your feedback will help us improve your experience
Christian Dell and 57 other Physics 103 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
When current $I$ flows through resistance $R$, the power generated is given by $W=I^{2} R$. Suppose that $I$ has a uniform distribution over the interval (0,1) and $R$ has a density function given by $$f(r)=\left\{\begin{array}{ll} 2 r, & 0 \leq r \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ Find the probability density function for $W$. (Assume that $I$ is independent of $R$.)
Functions of Random Variables
Summary
Let, $\varphi_{A}-\varphi_{B}=\varphi$ Now, thermal power generated in the resistance $R_{x}$ $$ P=i^{\prime 2} R_{x}=\left[\frac{\varphi}{R_{1}+\frac{R_{2} R_{x}}{R_{2}+R_{x}}} \frac{R_{2}}{R_{2}+R_{x}}\right]^{2} R_{x} $$ For $P$ to be independent of $R_{x}$ $$ \begin{aligned} &\frac{d P}{d R_{x}}=0, \text { which yeilds } \\ &R_{x}=\frac{R_{1} R_{2}}{R_{1}+R_{2}}=12 \Omega \end{aligned} $$
Electrodynamics
Electric Current
Use the WKB approximation to find the allowed energies of the general power-law potential: $$ V(x)=\alpha|x|^{\nu} . $$ where $v$ is a positive number. Check your result for the case $v=2 .$ Answer $^{14}$ $$ E_{n}=\alpha\left[(n-1 / 2) \hbar \sqrt{\frac{\pi}{2 m \alpha}} \frac{\Gamma\left(\frac{1}{v}+\frac{3}{2}\right)}{\Gamma\left(\frac{1}{v}+1\right)}\right]^{\left(\frac{2 v}{i+2}\right)} $$
The WKB Approximation
The Connection Formulas
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD