Classical Hills and Wells
CLASSICAL HILLS AND WELLS
Ian Cooper
School of Physics, The University of Sydney, NSW 2006, Australia
In introductory quantum mechanics courses, the Schrodinger equation is usually introduced and then applied to bound states and scattering by potential barriers. However, students encountering quantum mechanics for the first time are usually unfamiliar with the classical analogues used for bound states and barriers. This paper introduces a model and numerical procedures to describe the motion of a classical particle governed by potential energy “hills” and “wells” allowing many concepts in quantum mechanics such as potential energy, total energy, expectation values, uncertainty and probability to be studied. The investigation of the motion of a classical particle under the influence of potential hills and wells can be used as a starting point in introductory quantum mechanics. Students can then better grasp the ideas of classical physics and better appreciate fundamental ideas for a quantum description of a particle.
I. INTRODUCTION
Students commencing a quantum mechanics course for the first time are often exposed to the one dimensional Schrodinger equation being applied to (1) a particle trapped in a potential well and the characteristics of bound states are developed and (2) the scattering of a particle by a potential hill or well. The idea of a particle as a wave is a very complex one and the great number of new concepts is bewildering to many students and many misconceptions develop1,2,3. In introductory quantum mechanics courses, comparisons are continuously made between the predictions of quantum mechanics with those of classical mechanics. A problem is that students are not always familiar with what classical mechanics predicts and they have not studied the motion of a classical particle under the influence of a potential representing a hill or well.
This paper will introduce a model and numerical procedures to describe the motion of a classical particle that is governed by a potential energy function. From the model, the ideas of classical physics can be brought out to help students make a better comparison with the results predicted by quantum mechanics4. There are now many mathematical software packages such as MS EXCEL5, Matlab6, Mathematica7, Mathcad, and Maple available to teachers and students that can be used to model physical systems. All calculations for this paper were performed using Matlab and details of the numerical procedures are given in the appendix.
II. MOTION OF A CLASSICAL FREE PARTICLE
Consider a classical particle of mass m that can slide along a track of length L with no dissipative forces acting. The origin is taken as the centre of the track and the motion of the particle is only valid within the range from x = -L/2 to +L/2. The initial position of the particle is restricted to the range –L/2 £ x(0) 0. The “shape” of the track is determined by the potential energy function U(x). A convenient hill or well is a truncated parabola as given by Eq. (1). Um is the height of the hill (Um > 0) or depth of the well (Um < 0) and xm is the half-width of the hill or well,
(1)
The potential energy function is specified at N discrete points, xc, (c = 1, 2, 3, …, N) along the track. The force F(xc) acting on the particle at position xc is calculated from potential energy function using the difference approximation to the derivative, Eqns. (A1) and (A3). A potential hill (Um = + 100 J, xm = 6.00 m) and a potential well (Um = - 100 J, xm = 6.00 m) and the corresponding forces acting on the particle are shown in Fig. 1. For a particle within the range –xm < x < xm, the force graphs shown in Fig. 1. indicate that the a hill gives a repulsive force for a particle on either side of the apex whereas a there is an attractive force for the particle on either side of the bottom of the well.
The displacement of the particle as a function of time x(t) can be calculated directly from Newton’s second law, using the approximation for the second derivative, and the method outlined in the appendix. The initial conditions are specified by the position x(t = 0) and the kinetic energy K(t = 0). The initial kinetic energy is given rather than the initial velocity so that the initial kinetic energy is easily compared to the height or depth of the well Um. The calculation of the position x(t) given by Eq. (A5) is done until the particle moves out of the region -L/2 < x < +L/2 or t reaches some maximum value tmax. Once the position of the particle is known as a function of time, other dynamical quantities can be calculated as shown in Table 1.
force / F(t) = F(xc) where x(t) » xcpotential energy / U(t) = U(xc) where x(t) » xc
acceleration / a(t) = F(t) / m
velocity / v(t) = {x(t + Dt) – x(t - Dt) } / 2 Dt
kinetic energy / K(t) = 1/2 m v(t)2
total energy / E(t) = K(t) + U(t)
momentum / p(t) = m v(t)
Table 1. Dynamical quantities of the classical particle when located at position x(t) » xc.
Fig. 2. shows the graphs for x(t), v(t), a(t), K(t), U(t) and E(t) for a particle of mass m = 10.0 kg launched towards a potential hill (height Um = 100 J, L = 20.0 m and xm = 6.00 m) with an initial kinetic energy K(0) = 110 J and from a starting point x(0) = - 10.0 m. Initially the particle moves to the right until the repulsive force acts to slow the particle. Since the initial kinetic energy K(0) > Um, the particle can pass over the potential barrier. As the particle approaches the hill the potential energy increases at the expense of the kinetic energy. On passing the apex, the particle gains kinetic energy and loses potential energy. For the graphs shown in Fig. 3, the initial kinetic energy of the particle, K(0) = 90 J, is less than the height of the hill, Um = 100 J. The repulsive force acts to slow the particle, causing it to stop and reverse direction, therefore, the particle is reflected. Thus, the classical particle either passes over a barrier or is reflected by it and the particle’s total energy remains constant during the motion.
When the particle enters a potential well as shown in Fig. 4, the force is attractive and the particle speeds up and its kinetic energy increases while its potential energy decreases until it reaches the bottom of the well. At the bottom of the well, the force on the particle is zero but since it is moving at this position, the particle continues moving to the right. Now the force acting on the particle has reversed direction slowing the particle. Since the total energy of the particle is conserved, the particle always has sufficient kinetic energy to escape the well with a kinetic energy equal to its initial value.
III. MOTION OF A CLASSICAL BOUND PARTICLE
Consider a particle released at time t = 0 from a position to the left of the origin at x(0) = - 6.00 m in a potential well with parameters, Um = - 100 J, xm = 8.00 m and zero initial kinetic energy K(0) = 0. The force acting on the particle is always directed to the origin at x = 0. The motion for the particle is shown in Fig. 5. The graphs indicate that the particle executes simple harmonic motion and that the particle is bound to the region – 6.00 m £ x(t) £ + 6.00 m, that is, the particle can never escape from the well.
The potential energy function U(x) for the motion of the particle in Fig. 5. is
(2)
For a mass and spring system, the stored potential energy is U = 1/2kx2 where k is the spring constant and the period of oscillation is . For our classical particle, its effective spring constant is k = 200/82 N.m-1and the period of oscillation is T = 11.2 s which agrees with the value obtained graphically from Fig. 5. Using similar arguments it is possible to calculate the classical period, T, frequency, f, and angular frequency, w for a quantum particle in a harmonic well from the potential energy function. The energy level spectrum for a harmonic oscillator is
where n = 1, 2, 3, … . This is a discrete spectrum, whereas the total energy for our classical particle can have a continuous range of values depending upon the initial conditions. The variations of the kinetic, potential and total energies with time are shown in Fig. 5. The negative sign for the total energy implies that the particle is bound to the region – 6.00 m £ x(t) £ 6.00 m and at any time t, K(t) + U(t) = E .
IV. BOUND STATES
Newton’s second law and the Schrodinger wave equation have both been used with great success in explaining observed behaviour. Newton’s second law gives precise values of a system’s parameters, whereas a mathematical quantity known as the wave function y is used in the Schrodinger equation which gives results that can only be interpreted in terms of probabilities. In order to be useful, the wave mechanics formalism must predict values of measurable quantities like position, momentum and energy. This is done by using the wave function to calculate the expected result of the average of many measurements of a given quantity. The result is called the expectation value8. For motion in one dimension, the classical expectation value of a physical observable quantity Q(x) is
(3)
where P(x)dx is the probability distribution. This means that all values of Q(x) are each weighted by its frequency or probability of occurrence. For our bound classical particle we can also calculate expectation values, that is, we can find the average of a physical quantity q over one period T
(4)
The integral is approximated numerically. This is very easy to do in a spreadsheet where data are presented in columns. In Matlab, the trapz command can be used where data is stored in arrays, for example, the code to calculate the average value of kinetic energy <K> is
Kavg = trapz(t(cstart:cstop),K(cstart, cstop))/period
where ctstart and cstop are index numbers to evaluate the integral between values corresponding to one period. Table 2 summaries the expectation values for the parameters characterising the motion of the classical particle trapped in the well.
parameter / expectation valuekinetic energy / + 28.1 J
potential energy / - 71.9 J
total energy / - 43.8 J
position / 0 m
momentum / 0 kg.m.s-1
Table 2. Expectation values for a particle of mass 10.0 kg trapped in the well defined by Eq. (2). The period, T = 11.2 s is calculated from the index numbers cstart and cstop.
For the energy expectation values, <K> + <U> = <E>, that is, energy is conserved. The average position of the particle is zero because the amount of time spent in the region x < 0 is identical to the time spent in the region x > 0. The average momentum <p> is zero since the particle is either travelling to the left or to the right with equal probability.
Quantum mechanics associates an operator, [Q] with each observable Q and using an operator, one can calculate the average value <Q> of the corresponding observable by generalizing Eqn. (3)
In the wave mechanics approach, measurable quantities are related to probabilities therefore there is always some uncertainty in the measurement. The uncertainty Dq in measurement q may be found from the scatter of measurements about the average9, and the amount of scatter is measured by the standard deviation sq which is given by
(5)
For the particle trapped in the classical well, the uncertainty in position is Dx = 4.2 m and the uncertainty in momentum is Dp = 23.7 kg.m.s-1. The Heisenberg uncertainty principle in one dimension is defined as
(6)
and for our classical particle the uncertainty principle is well satisfied with Dx.Dp = 100 kg.m2.s-1.
In quantum mechanics the location of the particle is described in terms of the probability of locating the particle within some region. For our bound classical particle, oscillating back and forth, the probability of locating the particle within a given distance interval, Dxp is proportional to the time spent within this interval. Since Dt = Dxp / v, the probability density is inversely proportional to the velocity of the particle. The unscaled classical probability density function is shown in Fig. 6 where the velocity of the particle is found as a function of position for one period of the motion. The particle has maximum velocity at the centre of the well and as the particle approaches the extremes of the motion, the particle must slow down, stop and reverse direction. Therefore, the time intervals near the extremes are much larger than that near the centre of the well. This means that there is a much higher probability of locating the particle at the edges of the well than at its centre. This result is very different from the result predicted in quantum mechanics for the ground state of a harmonic oscillator, where there is maximum probability of locating the particle at the centre of the well and low probability at the edges.
The classical particle has a well defined trajectory and the dynamical values can be calculated at any time and the potential energy and force acting on the particle can be given at each position. This is very different to that of a quantum particle. In the Schrodinger equation, the potential energy term governs the motion, but it does not represent the potential energy of the particle at each position x. The concepts of a trajectory, position, velocity and acceleration are not defined. The location of the particle can only be expressed in terms of probabilities. Dynamics quantities such as kinetic energy, potential energy, total energy and momentum are given only as an average value and associated with this average value is an inherent uncertainty, they can not be expressed as a function of position of the particle.