A Fermi oscillator has Hamiltonian H=f^† f, where f is an operator that satisfies f^2=0, f f^†

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A Fermi oscillator has Hamiltonian H=f^† f, where f is an...

A Fermi oscillator has Hamiltonian $H=f^{\dagger} f$, where $f$ is an operator that satisfies $$ f^{2}=0, \quad f f^{\dagger}+f^{\dagger} f=1 $$ Show that $H^{2}=H$, and thus find the eigenvalues of $H$. If the ket $|0\rangle$ satisfies $H|0\rangle=0$ with $\langle 0 \mid 0\rangle=1$, what are the kets (a) $|a\rangle \equiv f|0\rangle$, and (b) $|b\rangle \equiv f^{\dagger}|0\rangle ?$ In quantum field theory the vacuum is pictured as an assembly of oscillators, one for each possible value of the momentum of each particle type. A boson is an excitation of a harmonic oscillator, while a fermion in an excitation of a Fermi oscillator. Explain the connection between the spectrum of $f^{\dagger} f$ and the Pauli principle.

A Fermi oscillator has Hamiltonian $H=f^{\dagger} f$, where $f$ is an operator that satisfies
$$
f^{2}=0, \quad f f^{\dagger}+f^{\dagger} f=1
$$
Show that $H^{2}=H$, and thus find the eigenvalues of $H$. If the ket $|0\rangle$ satisfies $H|0\rangle=0$ with $\langle 0 \mid 0\rangle=1$, what are the kets (a) $|a\rangle \equiv f|0\rangle$, and (b) $|b\rangle \equiv f^{\dagger}|0\rangle ?$
In quantum field theory the vacuum is pictured as an assembly of oscillators, one for each possible value of the momentum of each particle type. A boson is an excitation of a harmonic oscillator, while a fermion in an excitation of a Fermi oscillator. Explain the connection between the spectrum of $f^{\dagger} f$ and the Pauli principle.

Key Concepts

Fermi Oscillator

A Fermi oscillator is a quantum system characterized by fermionic operators that obey anticommutation relations. Its Hamiltonian is typically given as H = f† f, which essentially counts the presence or absence of a fermion in a particular state. This simple structure reflects the binary nature of fermionic occupation numbers, essential for modeling systems that obey Fermi-Dirac statistics.

Anticommutation Relations

Anticommutation relations are central to the behavior of fermionic operators. They are defined such that the square of the annihilation operator is zero (f^2 = 0) and the combination f f† + f† f equals one. These relations ensure that applying an annihilation operator twice yields zero and that the creation and annihilation operators produce only states with occupation numbers of 0 or 1, highlighting the discrete occupancy inherent in fermionic systems.

Projection Property of the Hamiltonian

The property H^2 = H for the Hamiltonian H = f† f indicates that H is a projection operator. This means its eigenvalues can only be 0 or 1, corresponding to the absence or presence of a fermion, respectively. This mathematical structure simplifies analysis and directly reflects the physical constraint that a fermionic state cannot be multiply occupied.

Ladder Operators and Fock Space

In the context of Fock space, the ladder operators, f and f†, are used to lower and raise the fermion number of a given state, respectively. Starting from a vacuum state with no particles, the creation operator f† generates a single occupied state, while the annihilation operator f either reduces the occupancy or annihilates the state if already unoccupied. The anticommutation properties ensure that only two states, corresponding to 0 or 1 fermion, exist in each mode.

Pauli Exclusion Principle

The Pauli exclusion principle states that no two fermions can occupy the same quantum state simultaneously. The spectral property of the operator f† f, which only admits eigenvalues 0 and 1, is a direct mathematical manifestation of this principle. It ensures that each mode of a fermionic system can only be unoccupied or singly occupied, thereby preventing the possibility of multiple fermions sharing the same set of quantum numbers.

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