Spectrum of a Free Particle
Classical Mechanics
Level
2
What is the spectrum of the free-particle Hamiltonian (redefined up to a constant)
The set of real numbers
The set of perfect squares
The set of non-negative integers
The set of non-negative real numbers
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1 solution
The eigenfunctions of this Hamiltonian are the plane waves e i k x ; each wavenumber k corresponds to the eigenvalue k 2 , a positive real number. There also exists exactly a zero energy solution ψ 0 ∝ 1 , which has no degeneracy and eigenvalue 0 . Note that none of the free-particle states are normalizable, but this is not important for this question -- free particle states are only approximations in QM. The full spectrum of eigenvalues is thus the non-negative reals.