Question 1 · Física II URV 2024

Tema: Work, mechanical energy and potential energy curves · Del examen Exámenes de Física II: Teoría y Laboratorio (Mayo 2024) (2024)

Enunciado

Question 1 (5 points). Morse potential

A widely used model for describing the vibrational motion of diatomic molecules is the Morse potential. In a specific dimensionless one-dimensional formulation, the potential energy is given by:

V(x)=e−2x−2e−xV(x) = e^{-2x} - 2e^{-x}

Analyze and discuss the dynamics of a particle moving in this potential field if its total mechanical energy is E=−0.5 JE = -0.5\text{ J}.

1.

Position xmin⁡x_{\min} and value Vmin⁡V_{\min} of the potential energy minimum:

2.

For E=−0.5 JE = -0.5\text{ J}, calculate the inner (x1x_1) and outer (x2x_2) classical turning points, and the maximum kinetic energy Kmax⁡K_{\max}:

3.

Characterize the nature of the motion for E=−0.5 JE = -0.5\text{ J} (e.g. bounded vs unbounded, type of motion):

Enunciado: Question 1 · Física II URV 2024

Solución

Según los criterios oficiales de corrección

1. Position of equilibrium xmin⁡x_{\min}:

Solución: 00 m

1. Minimum potential energy Vmin⁡V_{\min}:

Solución: −1-1 J

2. Inner turning point x1x_1:

Solución: −0.535-0.535 m / −ln⁡(1+22)-\ln(1 + \frac{\sqrt{2}}{2}) m

2. Outer turning point x2x_2:

Solución: 1.2281.228 m / +1.228+1.228 m / −ln⁡(1−22)-\ln(1 - \frac{\sqrt{2}}{2}) m

2. Maximum kinetic energy Kmax⁡K_{\max}:

Solución: 0.50.5 J

3. Characterize the nature of the motion for E=−0.5 JE = -0.5\text{ J}:

Solución: Since Vmin⁡=−1<E<0V_{\min} = -1 < E < 0, the motion is bounded (a bound state). The particle undergoes periodic non-linear oscillations between the turning points x1≈−0.535 mx_1 \approx -0.535\text{ m} and x2≈1.228 mx_2 \approx 1.228\text{ m}.

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