Question

8. Two-dimensional free Fermion gas. In class we looked at a 3-D gas of non-interacting free Fermions. Here we will look at a 2-D Fermion gas in a plane of width $L$, which approximately describes a 2-D electron gas (e.g., in the quantum Hall and fractional quantum Hall effects). Let $m^*$ or $m$ represent the effective mass of each particle (be careful to avoid confusion if $m$ is used as an integer index). a. Employ periodic boundary conditions for a \"toroid,\" i.e. $\psi_k(x+L) = \psi_k(x)$ and $\psi_k(y+L) = \psi_k(y)$, where the plane wave eigenstates are given by: $\psi_k(x,y) = A \exp[i(k_xx+k_yy)]$. Determine the normalization factor $A$ and the allowed values of $k_x$ and $k_y$ (the latter in terms of quantum numbers $n_x$ and $n_y$, and the width $L$). b. What is the energy (energy eigenvalue) of each Fermion in terms of its effective mass and wavenumber: $k = \sqrt{k_x^2 + k_y^2}$?

          8. Two-dimensional free Fermion gas. In class we looked at a 3-D gas of non-interacting free Fermions.
Here we will look at a 2-D Fermion gas in a plane of width $L$, which approximately describes a 2-D
electron gas (e.g., in the quantum Hall and fractional quantum Hall effects). Let $m^*$ or $m$ represent the
effective mass of each particle (be careful to avoid confusion if $m$ is used as an integer index).
a. Employ periodic boundary conditions for a \"toroid,\" i.e. $\psi_k(x+L) = \psi_k(x)$ and $\psi_k(y+L) = \psi_k(y)$,
where the plane wave eigenstates are given by: $\psi_k(x,y) = A \exp[i(k_xx+k_yy)]$. Determine the
normalization factor $A$ and the allowed values of $k_x$ and $k_y$ (the latter in terms of quantum numbers $n_x$
and $n_y$, and the width $L$).
b. What is the energy (energy eigenvalue) of each Fermion in terms of its effective mass and
wavenumber: $k = \sqrt{k_x^2 + k_y^2}$?

$8. Two-dimensional free Fermion gas. In class we looked at a 3-D gas of non-interacting free Fermions. Here we will look at a 2-D Fermion gas in a plane of width L, which approximately describes a 2-D electron gas (e.g., in the quantum Hall and fractional quantum Hall effects). Let m^* or m represent the effective mass of each particle (be careful to avoid confusion if m is used as an integer index). a. Employ periodic boundary conditions for a ẗoroid,ï.e. (x+L) = (x) and (y+L) = (y), where the plane wave eigenstates are given by: (x,y) = A [i(kxx+kyy)]. Determine the normalization factor A and the allowed values of kx and ky (the latter in terms of quantum numbers nx and ny, and the width L). b. What is the energy (energy eigenvalue) of each Fermion in terms of its effective mass and wavenumber: k = √(kx^2 + ky^2)?$

Added by Susan I.

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