- #1
haziq
- 3
- 1
- Homework Statement
- Problem 2 in Chapter 2 of Hall’s QM book. See pictures below
- Relevant Equations
- See photo below
I’ve been trying to solve this for ages. Would really appreciate some hints. Thanks
Last edited:
haziq said:Homework Statement:: Problem 2 in Chapter 2 of Hall’s QM book. See pictures below
Relevant Equations:: See photo below
solve this for ages
BvU said:PF guidelines require that you actually post your best attempt before we are allowed to assist .
The general approach involves using the classical mechanics framework where the travel time can be computed by integrating the inverse of the particle's velocity over the path of motion. This typically requires expressing the velocity as a function of position and then performing the integral.
The velocity of a particle can be expressed using the conservation of energy principle. For a particle of mass \(m\) moving in a potential \(V(x)\), the total energy \(E\) is conserved and can be written as \(E = \frac{1}{2}mv^2 + V(x)\). Solving for the velocity \(v\), we get \(v = \sqrt{\frac{2}{m}(E - V(x))}\).
The travel time \(T\) is found by integrating the inverse of the velocity over the path of the particle. This can be written as \(T = \int_{x_1}^{x_2} \frac{dx}{v(x)}\), where \(v(x)\) is the velocity as a function of position. Substituting the expression for \(v(x)\), we get \(T = \int_{x_1}^{x_2} \frac{dx}{\sqrt{\frac{2}{m}(E - V(x))}}\).
The limits of integration \(x_1\) and \(x_2\) correspond to the initial and final positions of the particle along its path. These positions are typically determined by the points where the particle's kinetic energy is zero, i.e., where \(E = V(x)\), which are the turning points of the motion.
The potential energy function \(V(x)\) must be known or specified as part of the problem. Depending on its form, the integral may need to be evaluated analytically or numerically. For simple potentials like a harmonic oscillator or a constant potential, the integral can often be solved analytically. For more complex potentials, numerical methods may be necessary to evaluate the integral.