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Chapter 6: Problem 10

Calculate the first-order correction to the energy of a ground-state harmonic oscillator subject to an anharmonic potential of the form \(a x^{3}+b x^{4}\) where \(a\) and \(b\) are small (anharmonicity) constants. Consider the three cases in which the anharmonic perturbation is present (a) during bond expansion \((x \geq 0)\) and compression \((x \leq 0)\) (b) during expansion only, (c) during compression only.

Short Answer

Expert verified
In all the three cases (a) during bond expansion and compression, (b) during expansion only, (c) during compression only, the first-order quantum corrections to the energy can be calculated using the time independent perturbation theory equations and the ground-state wave functions.

Step by step solution

01

Identify the Ground State Wave Function and Energy

The ground state wave-function of the harmonic oscillator is given by \(\psi_{0}(x)=\left(\frac{m w}{\pi \hbar}\right)^{1 / 4} \exp \left(-\frac{1}{2} \frac{m w}{\hbar} x^{2}\right)\) and the energy is \(E_{0}=\frac{1}{2} \hbar w\). Recall that \(\hbar\) is the reduced Planck's constant, \(m\) is the mass, and \(w\) is the angular frequency.
02

Calculate First-Order Correction for Different Cases

The first-order correction to the energy in time-independent perturbation theory is given by \(E_{1}=\left\langle\Psi_{0}\right|H_{p}\left|\Psi_{0}\right\rangle\). Apply the Hamiltonian operator to the ground state wave-function, keeping only terms up to first order as we ignore higher-order corrections due to small anharmonic constants \(a\) and \(b\). The Hamiltonian is \(H_{p}=a x^{3}+b x^{4}\). For the three cases, calculate the energy correction by integrating over the appropriate range: (a) from \(-\infty\) to \(\infty\), (b) from 0 to \(\infty\), and (c) from \(-\infty\) to 0.
03

Evaluate the Integrals Using Gaussian Integral Properties

Evaluating any of these integrals can be made easier by using the properties of the Gaussian integral. We know that \(\int e^{-p x^{2}} d x=\sqrt{\pi / p}\), \(\int x^{2} e^{-p x^{2}} d x=\frac{1}{2 p} \sqrt{\pi / p}\) and \(\int x^{4} e^{-p x^{2}} d x=\frac{3}{4 p^{2}} \sqrt{\pi / p}\). For the case of expansion only or compression only, the correct form of integral must include a factor of 2, since we are only considering half of the \(x\) range.
04

Express the Final Energy

The final energy of the oscillator under an anharmonic potential after first order correction is the sum of the basic energy and the first-order correction, \(E_{\text{total}}=E_{0}+E_{1}\). Evaluate the integrals calculated in step 3 and add the result to the energy in the ground-state to obtain the corrected energy for the respective cases.

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Most popular questions from this chapter

Find the complete dependence of the \(A\) and \(B\) coefficients on atomic number for the \(2 \mathrm{p} \rightarrow 1\) s transitions of hydrogenic atoms. Calculate how the stimulated emission rate depends on \(Z\) when the atom is exposed to black-body radiation at \(1000 \mathrm{K} .\) Hint. The relevant density of states also depends on \(Z\).

Calculate the second-order energy correction to the ground state of a particle in a one-dimensional box for a perturbation of the form \(H^{(1)}=-\varepsilon \sin (\pi x / L)\) by using the closure approximation. Infer a value of \(\Delta E\) by comparison with the numerical calculation in Example \(6.4 .\) These two problems \((6.14 \text { and } 6.15)\) show that the parameter \(\Delta E\) depends on the perturbation and is not simply a characteristic of the system itself.

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.

A simple calculation of the energy of the helium atom supposes that each electron occupies the same hydrogenic 1 s-orbital (but with \(Z=2\) ). The electronelectron interaction is regarded as a perturbation, and calculation gives $$\int \psi_{1 s}^{2}\left(r_{1}\right)\left(\frac{e^{2}}{4 \pi \varepsilon_{0} r_{12}}\right) \psi_{1 s}^{2}\left(r_{2}\right) \mathrm{d} \tau=\frac{5}{4}\left(\frac{e^{2}}{4 \pi \varepsilon_{0} a_{0}}\right)$$ (see Example 7.2 ). Estimate (a) the binding energy of helium, (b) its first ionization energy. Hint. Use eqn 6.15 with \(E_{1}=E_{2}=E_{1 \mathrm{s}} .\) Be careful not to count the electronelectron interaction energy twice.

The symmetry of the ground electronic state of the water molecule is \(\mathrm{A}_{1}\). (a) An electric field, (b) a magnetic field is applied perpendicular to the molecular plane. What symmetry species of excited states may be mixed into the ground state by the perturbations? Hint. The electric interaction has the form \(H^{(1)}=a x ;\) the magnetic interaction has the form \(H^{(1)}=b l_{x^{*}}\)
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