(a) Write down the Hamiltonian for two noninteracting...

(a) Write down the Hamiltonian for two noninteracting identical particles in the infinite square well. Verify that the fermion ground state given in Example $5.1$ is an eigenfunction of $H$, with the appropriate eigenvalue. (b) Find the next two excited states (beyond the ones in Example 5.1) - wave functions and energies - for each of the three cases (distinguishable, identical bosons, identical fermions).

(a) Write down the Hamiltonian for two noninteracting identical particles in the infinite square well. Verify that the fermion ground state given in Example $5.1$ is an eigenfunction of $H$, with the appropriate eigenvalue.
(b) Find the next two excited states (beyond the ones in Example 5.1) - wave functions and energies - for each of the three cases (distinguishable, identical bosons, identical fermions).

Key Concepts

Hamiltonian Operator

The Hamiltonian operator in quantum mechanics represents the total energy (kinetic plus potential) of a system. For noninteracting particles, the Hamiltonian is separable into individual parts that each act on the degrees of freedom of a single particle, which simplifies finding the eigenfunctions and eigenvalues.

Infinite Square Well

The infinite square well is a fundamental model in quantum mechanics where the potential is zero inside a finite region and infinite outside. This leads to quantized energy levels and wave functions that must vanish at the boundaries of the well, forming a complete set of orthonormal eigenfunctions.

Identical Particles

Identical particles are those that are indistinguishable by their intrinsic properties. In quantum mechanics, they require special treatment through symmetrization or antisymmetrization of the wavefunctions, depending on whether the particles are bosons or fermions.

Symmetrization Postulate

The symmetrization postulate dictates that the wave function of a system of identical particles must be symmetric for bosons (unchanged under particle exchange) and antisymmetric for fermions (changes sign under exchange). This has significant implications for the allowed energy states and the structure of multi-particle wavefunctions.

Energy Eigenstates and Eigenvalues

Energy eigenstates are solutions to the time-independent Schrödinger equation that correspond to definite energy values (eigenvalues). For systems like the infinite square well, these eigenstates are well-defined and quantized, and they form the basis for constructing multi-particle states.

Noninteracting Particles

In the context of noninteracting particles, the total Hamiltonian is just the sum of the individual Hamiltonians of each particle. This independence allows the multi-particle wavefunctions to be constructed from products or symmetrized/antisymmetrized combinations of single-particle eigenfunctions, facilitating the analysis of states with multiple particles.

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