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Chapter 19: Problem 19

Show that the Morse potential approaches the harmonic potential for small values of the vibrational amplitude. (Hint: Expand the Morse potential in A Taylor-Maclaurin series.)

Short Answer

Expert verified
The Morse potential can be approximated by the harmonic potential for small values of the vibrational amplitude by expanding the Morse potential as a Taylor-Maclaurin series around the minimum point of the potential curve: \(V(x) \approx D_e\alpha^2(x - x_e)^2\). Comparing this expansion to the harmonic potential \(V_{harm}(x) = \frac{1}{2}k(x - x_e)^2\), we find that the Morse potential approaches the harmonic potential when vibrational amplitude is small, as the term \((x - x_e)^2\) will also be small.

Step by step solution

01

Taylor-Maclaurin series expansion of the Morse potential

To expand the Morse potential as a Taylor-Maclaurin series around the point \(x = x_e\), we have the Taylor-Maclaurin series formula given by: \[f(x) =f(x_e) + f'(x_e)(x - x_e) + \frac{1}{2} f''(x_e) (x - x_e)^2 + \dots \] To evaluate the first few derivatives of V(x) at x = x_e: \(V(x) = D_e [1 - e^{-\alpha(x - x_e)}]^2 = D_e(1 - e^{-\alpha(x - x_e)})^2\) Differentiating with respect to x: \(V'(x) = 2D_e\alpha e^{-\alpha(x - x_e)} (1 - e^{-\alpha(x - x_e)})\) \(V''(x) = 2D_e\alpha^2 e^{-\alpha(x - x_e)}(3 - 4e^{-\alpha(x - x_e)} + e^{-2\alpha(x - x_e)})\) Now we evaluate:\(V(x_e), V'(x_e)\) and \(V''(x_e)\). \(V(x_e) = D_e [1 - e^{-\alpha(x_e - x_e)}]^2 = D_e(1 - 1)^2 = 0\) \(V'(x_e) = 2D_e\alpha e^{-\alpha(x_e - x_e)} (1 - e^{-\alpha(x_e - x_e)}) = 2D_e\alpha(1 - 1) = 0\) \(V''(x_e) = 2D_e\alpha^2 e^{-\alpha(x_e - x_e)}(3 - 4e^{-\alpha(x_e - x_e)} + e^{-2\alpha(x_e - x_e)}) = 2D_e\alpha^2(3 - 4 + 1) = 2D_e\alpha^2\) Now we plug these values into the Taylor-Maclaurin series: \[V(x) \approx 0 + 0(x - x_e) + \frac{1}{2}(2D_e\alpha^2) (x - x_e)^2\] \[V(x) \approx D_e\alpha^2(x - x_e)^2\]
02

Compare the expansion to the harmonic potential

Now, we compare the expansion of the Morse potential with the given harmonic potential: \[V_{harm}(x) = \frac{1}{2}k(x - x_e)^2\]
03

Analyze when the Morse potential approaches the harmonic potential

Comparing the expressions, we see that the Morse potential approaches the harmonic potential for small values of the vibrational amplitude when: \[D_e\alpha^2(x - x_e)^2 \approx \frac{1}{2}k(x - x_e)^2\] Notice that if the vibrational amplitude (which corresponds to the difference between x and \(x_e\)) is small, then the term \((x - x_e)^2\) will also be small. Therefore, we can approximate the Morse potential by the harmonic potential in this case, as requested.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Harmonic Potential
The harmonic potential is a fundamental concept in physics, particularly when discussing oscillatory motion. It describes the potential energy of a system that exhibits simple harmonic motion, such as a mass connected to a spring. Mathematically, the harmonic potential is expressed as:
\[ V_{harm}(x) = \frac{1}{2} k (x - x_e)^2 \]
where:
  • \( k \) is the spring constant, a measure of the stiffness of the spring.
  • \( x \) is the displacement of the object from equilibrium.
  • \( x_e \) is the equilibrium position.
  • The equation is characterized by its quadratic form, which ensures that the restoring force is always proportional to the displacement.
This quadratic expression signifies that the potential energy increases with the square of the amplitude of displacement, directly leading to Hooke's Law. When dealing with small oscillations or small vibrational amplitudes, the approximation of harmonic potential is very useful, simplifying more complex potentials like the Morse potential.
Taylor-Maclaurin Series
The Taylor-Maclaurin series is a tool used to approximate functions. It's particularly helpful when simplifying complex functions into more manageable forms. For an analytical function \( f(x) \) around a point \( x = x_e \), the Taylor-Maclaurin series expansion is:
\[ f(x) = f(x_e) + f'(x_e)(x - x_e) + \frac{1}{2} f''(x_e) (x - x_e)^2 + \dots \]
This expansion portrays the function as an infinite sum of its derivatives at a specific point, making it easier to approximate. The idea is to take the value of the function and its successive derivatives at \( x = x_e \) and construct a polynomial that approximates \( f(x) \).
  • The first derivative term \( f'(x_e) \) approximates the slope or rate of change of the function at the point.
  • The second derivative term \( \frac{1}{2} f''(x_e) (x - x_e)^2 \) considers the curvature or how the slope itself changes.
When applied to the Morse potential, such an expansion helps in revealing its similarity to the harmonic potential for small vibrational amplitudes.
Vibrational Amplitude
Vibrational amplitude refers to the extent of displacement in a vibrational system from its equilibrium position. It is often a key parameter in understanding oscillatory systems like molecules and springs.
In the context of the Morse potential and harmonic potential comparison, small vibrational amplitudes imply the displacement, \( x - x_e \), is small. Because of this, higher-order terms in a series expansion, such as the Taylor-Maclaurin series, become negligible.
  • This results in a simplified view, where only the second order, or quadratic term, heavily contributes to the potential.
  • In practical terms, this approximation is very useful for molecular vibrations and places where harmonic oscillators approximate closely how particles behave.
Therefore, when vibrational amplitude is small, the Morse potential which usually accounts for anharmonicities (deviations from Hooke's Law), effectively behaves like the harmonic potential. This conceptual bridge is crucial for understanding molecular dynamics in simplified terms.

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Most popular questions from this chapter

The fundamental vibrational frequencies for \(^{1} \mathrm{H}_{2}\) and \(^{2} \mathrm{D}_{2}\) are 4401 and \(3115 \mathrm{cm}^{-1},\) respectively, and \(D_{e}\) for both molecules is \(7.677 \times 10^{-19} \mathrm{J}\). Using this information, calculate the bond energy of both molecules.

A strong absorption band in the infrared region of the electromagnetic spectrum is observed at \(\tilde{\nu}=1298 \mathrm{cm}^{-1}\) for \(^{40} \mathrm{Ca}^{1} \mathrm{H}\). Assuming that the harmonic potential applies, calculate the fundamental frequency \(\nu\) in units of inverse seconds, the vibrational period in seconds, and the zero point energy for the molecule in joules and electron-volts.

The rotational constant for \(^{7} \mathrm{Li}^{19} \mathrm{F}\) determined from microwave spectroscopy is \(1.342583 \mathrm{cm}^{-1} .\) The atomic masses of \(^{7} \mathrm{Li}\) and \(^{19} \mathrm{F}\) are 7.00160041 and 18.9984032 amu, respectively. Calculate the bond length in \(^{7} \mathrm{Li}^{19} \mathrm{F}\) to the maximum number of significant figures consistent with this information.

The rotational constant for \(^{14} \mathrm{N}_{2}\) determined from microwave spectroscopy is \(1.99824 \mathrm{cm}^{-1}\). The atomic mass of \(^{14} \mathrm{N}\) is 14.003074007 amu. Calculate the bond length in \(^{14} \mathrm{N}_{2}\) to the maximum number of significant figures consistent with this information.

Purification of water for drinking using UV light is a viable way to provide potable water in many areas of the world. Experimentally, the decrease in UV light of wavelength \(250 \mathrm{nm}\) follows the empirical relation \(I / I_{0}=e^{-s^{\prime} l}\) where \(l\) is the distance that the light passed through the water and \(\varepsilon^{\prime}\) is an effective absorption coefficient. \(\varepsilon^{\prime}=0.070 \mathrm{cm}^{-1}\) for pure water and \(0.30 \mathrm{cm}^{-1}\) for water exiting a wastewater treatment plant. What distance corresponds to a decrease in \(I\) of \(15 \%\) from its incident value for (a) pure water and (b) waste water?
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