Anharmonic Oscillator

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A anharmonic oscillator is characterized by potential $V (x) = \frac{m ω^{2}}{2} x^{2} + λ x^{4}$ where $λ$ is a positive constant. With $λ = 0$ the potential of the harmonic oscillator is obtained. It is not possible to analytically solve the Schrödinger equation. Unless otherwise specified we have set the following values: $ℏ = 1$ ; $m = 1$ ; $ω = 1$ . Each graph displays the harmonic potential $\frac{m ω^{2}}{2} x^{2}$ (in blue color) and the quadratic potential $λ x^{4}$ (in red color).
	The Mathematica notebook, given a potential V(x), allows to numerically solve the equation, finding the energies (eigenvalues) and the relative functions (eigenfunctions).
EIGENVALUES

$λ$ =0 versus $λ$ =0.01 The total potential is almost harmonic (the harmonic component is the blue line). The comparison shows a small difference in the eigenvalues.

$λ$ =0 versus $λ$ =0.1 The value of the total potential depends on both the harmonic component (blue color) and the quadratic component (red color). The deviation of the eigenvalues is still small.

$λ$ =0 versus $λ$ =1 The potential is nearly quadratic (red color). The deviation of the eigenvalues is significant.
EIGENVALUES and EIGENFUNCTIONS

$λ$ =0 versus $λ$ =0.01 The total potential is almost harmonic (the harmonic component is the blue line). The comparison shows a small difference in the eigenvalues.

$λ$ =0 versus $λ$ =0.1 The value of the total potential depends on both the harmonic component (blue color) and the quadratic component (red color). The deviation of the eigenvalues is still small.

$λ$ =0 versus $λ$ =1 The potential is nearly quadratic (red color). The deviation of the eigenvalues is significant.