Homework Set 3

Problem 1: Canonical partition function for two noninteracting fermions in a box

Consider two spinless non-interacting fermions enclosed in a box of volume Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V } . Each fermion is described by a single particle state Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle | \mathbf{k}_i \rangle } , whose coordinate representation is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle \mathbf{r} | \mathbf{k}_i \rangle } .

(a) Using single particle states and , construct the two-particle quantum state that obeys the antisymmetrization postulate of quantum mechanics. (b) Using from (a), evaluate the canonical partition function for this system:

     Here you will find the useful the following integral:
                                    

(c) The partition function from (b) can be cast into the following form:

     What is the physical meaning of the first term and the second term in this expression?   Here  is the thermal de Broglie wavelength.
     (d)  Using your result in (a) and (b), find the density operator for these two fermions.

Problem 2: Electrons in graphene

Graphene is one-atom-thick crystal of carbon atoms densely packed into a honeycomb lattice. Its surprising discovery (according to Mermin-Wagner theorem of statistical mechanics, two-dimensional crystals do not exist!) in 2004 has led to Nobel Prize in Physics 2010. The band structure of graphene close to the Fermi energy is such that electrons behave as the so-called massless Dirac fermions with energy-momentum relationship:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon_{\pm} (\mathbf{k}) = \pm \hbar v_F |\mathbf{k}| }

akin to photons or neutrinos of high energy physics, except that instead of the velocity of light Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c } is replaced by the Fermi velocity is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_F \approx c/300 } .

(a) For any fermionic system at chemical potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu } , show that the probability of finding an occupied state of energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu + \delta } is the same as that of finding an unoccupied state of energy $\mu -\delta$ where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta } is any constant energy. HINT: According to Fermi-Dirac statistics, the probability of finding Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_\varepsilon } particles in a single particle state of energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon } is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P [n_\varepsilon] = \frac{e^{-\beta(\varepsilon - \mu)n_\varepsilon}}{1+e^{-\beta(\varepsilon - \mu)}} }

(b) At zero temperature, all negative energy states (the so-called holes) in graphene are occupied and all positive energy states are empty, so that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu (T=0) =0 } . Using the result in (a), find the chemical potential at finite temperature Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T } .

(c) Show that the mean excitation energy of this system at finite temperature satisfies:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(T) - E(0) = 4 A \int \frac{d^2 \mathbf{k}}{(2 \pi)^2} \frac{\varepsilon_+(\mathbf{k})}{\exp[\beta \varepsilon_+(\mathbf{k})] + 1 } }

where A is the surface are of graphene.

(d) Give a closed form answer for the excitation energy by evaluating integral in (c).

(e) Using (d), calculate the heat capacity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_V } of massless Dirac fermions in graphene as a function of temperature.

Note that your final results can be expressed using Riemann zeta function which specifies the values of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_m^-(z) } for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z=e^{\beta \mu}=1 } in the case of Dirac fermions.

Problem 3: Pauli paramagnetism

Calculate the contribution of electron spin to its magnetic susceptibility as follows. Consider non-interacting electrons where each is subject to a Hamiltonian operator:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H}_1 = \frac{\hat{\mathbf{p}}^2}{2m} - \mu_0 \vec{\sigma} \cdot \mathbf{B} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_0 } is the Bohr magneton and we ignore orbital effects of magnetic field (if they are taken into account then, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} \rightarrow \mathbf{p} - e\mathbf{A} } for vector potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} } ).

(a) Calculate the grand potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Phi } at a chemical potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu } . HINT: The energy of electron gas is given by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E = \sum_p E_p (n_p^+,n_p^-) } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_p^\pm } (=0 or 1 as in the case of any fermionic system) denotes the number of particles having Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm } spins and momentum p, and

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_p(n_p^+,n_p^-) = \left(\frac{p^2}{2m} - \mu_0 B \right) n_p^+ + \left(\frac{p^2}{2m} + \mu_0 B \right) n_p^- = (n_p^+ + n_p^-) \frac{p^2}{2m} - (n_p^+ - n_p^-) \mu_0 B } .

(b) Find the densities Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_+ = N_+ /V } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_- = N_- /V } of electrons pointing parallel and antiparallel to the magnetic field, respectively.

(c) Using result in (b), find the magnetization Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M = \mu_0 (N_+ - N_-) } , and expand the result for small B.

(d) Sketch the zero-field susceptibility Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \chi(T)/N = \partial M/\partial B |_{B=0}/N } per particle, and indicate its behavior at low and high temperatures. HINT: In the low temperature limit Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \ln z=\beta \mu \rightarrow \infty} you will find useful the following identity:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_n^-(z) \approx \frac{[\ln(z)]^n}{n\Gamma(n)} } .

(e) Estimate the magnitude of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \chi(T)/N } for a typical metal at room temperature. HINT: Since room temperature is always smaller that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_F \sim 10^4 } K of typical metals, you can take low temperature limit Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T \rightarrow 0 } of your result in (d).

Problem 4: Stoner ferromagnetism

The conduction electrons in a metal can be treated as a gas of fermions of spin 1/2 and density Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=N/V } . The Coulomb repulsion favors wave functions that are antisymmetric in position coordinates, thus keeping the electrons apart. Because of the full (position and spin) anti-symmetry of fermionic wave functions, this interaction may be approximated by an effective spin-spin coupling that favors states with parallel spins. In this simple approximation, the net effect is described by an interaction energy:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U = \alpha \frac{N_+N_-}{V} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_+ } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_- = N - N_+ } are the numbers of electrons with up and down spins, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V } is the volume. The parameter $\alpha$ is related to the scattering length Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a } by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha = 4 \pi \hbar^2 a/m } .

(a) The ground state has two Fermi seas filled with spin-up and spin-down electrons. Express the corresponding Fermi wavevector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{F\pm} } in terms of densities Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_\pm = N_\pm /V } .

(b) Calculate the kinetic energy density Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_{\mathrm{kin}} } of the ground state as a function of densities $n_{\pm }$ and the fundamental constants.

(c) Assuming small deviations Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_\pm = n/2 \pm \delta } from the symmetric state, expand the kinetic energy to fourth order in Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta } .

(d) Express the spin-spin interaction density Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U/V } in terms of $n$ and $\delta$. Find the critical value of $\alpha _{c}$ such that $\alpha >\alpha _{c}$ the electron gas can lower its total energy (which is the sum $E_{\mathrm {total} }/V=E_{\mathrm {kin} }/V+U/V$ of kinetic and potential energy) by spontaneously developing a magnetization. That is, when $\alpha >\alpha _{c}$, the total energy density $E_{\mathrm {total} }/V$ becomes lower due to negative contribution of the terms multiplying Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta^2 } . Such favorable finite Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta } means non-zero magnetization $M=\mu _{0}(N_{+}-N_{-})$ which in this case, unlike in problem 2., is spontaneous since it occurs in the absence of external magnetic field. This is known as the Stoner instability.

(e) Explain qualitatively the behavior of spontaneous magnetization as a function of $\alpha$ and sketch the corresponding graph.

Problem 5: Neutron stars

A neutron star can be considered to be a collection of non-interacting neutrons, which are Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{spin-}\frac{1}{2}} fermions. A typical neutron star has a mass Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M } close to one solar mass Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_\bigodot \approx 2 \times 10^{30} } kg. The mass of a neutron is about Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m = 1.67 \times 10^{-27} } kg. In the following, we will estimate the radius $R$ of the neutron star.

(a) Find the energy of a neutron star at $T=0$ as a function of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R } , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M } , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m } assuming that the star can be treated as an ideal non-relativistic Fermi gas.

(b) Assume that the density of the star is uniform and show that its gravitational energy is given by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_G = -3 G M^2/5R } , where the gravitational constant $G=6.67\times 10^{-11}Nm^{2}/kg^{2}$. HINT: From classical mechanics find the gravitational potential energy between an existing sphere of radius Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r } and a shell of volume Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4\pi r^2 dr } coming from infinity to radius Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r } . Then integrate that expression from Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=0 } to $R$.

(c) Assume that gravitational equilibrium occurs when the total energy is minimized and find an expression for the radius $R$.

(d) Estimate the actual value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R } in kilometers. Estimate the mass density and compare it with the density of material on the surface of Earth such as water.

(e) Determine the Fermi energy and Fermi temperature of a neutron star. A typical internal temperature for a neutron star is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T=10^8 } K. Compare this value with the Fermi temperature and determine if the zero temperature approximation that we assumed is applicable.

(f) Compare the rest energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle mc^2 } of a neutron star with Fermi energy of a neutron star. Is the non-relativistic approximation justified?