Classical Hills and Wells
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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
1 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,
x > L / 2 U(x) isnotdefined (1)
> U(x ) = 0 x xm
ж 2ц з ж ц ч x Ј x ( ) = 1 - з x ч m U x Um з з ч ч и иx m ш ш
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) xc
potential energy U(t) = U(xc) where x(t) xc
acceleration a(t) = F(t) / m
velocity v(t) = {x(t + t) – x(t - t) } / 2 t
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.
2 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 100 ( ) = -100+ 2. x Ј 8m U x 2 x (2) 8 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 T = 2p m . For our classical particle, k its effective spring constant is k = 200/82 N.m-1 and 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, for a quantum particle in a harmonic well from the potential energy 1 function. The energy level spectrum for a harmonic oscillator is En = (n - 2 )hw 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
3 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 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 Q (x )P(x )dx * < Q >= т P (x )dx = y (x ) y (x )dx , (3) P(x )dx т 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 t +T q dt (4) < q >= тt . T
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
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 value kinetic 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.
4 For the energy expectation values, 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 * Y (x )[Q(x )] Y(x ) dx < Q >= т Ч Y (x )* Y(x ) dx т In the wave mechanics approach, measurable quantities are related to probabilities therefore there is always some uncertainty in the measurement. The uncertainty q 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 q which is given by s = Dq = < q 2 > - < q >2 . q (5) For the particle trapped in the classical well, the uncertainty in position is x = 4.2 m and the uncertainty in momentum is p = 23.7 kg.m.s-1. The Heisenberg uncertainty principle in one dimension is defined as h Dx ЧDp і , (6) 2 and for our classical particle the uncertainty principle is well satisfied with x.p = 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, xp is proportional to the time spent within this interval. Since t = xp / 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 5 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. VI. DISCUSSION In this paper it has been demonstrated that many concepts in quantum mechanics such as potential energy, total energy, expectation values, uncertainties and probability can be applied to the motion of a classical particle. Using high speed personal computers and mathematical software it is easy to implement numerical methods to study a wide range of physical systems such as described in this paper, that is, the motion of a classical particle governed by a truncated parabolic potential. The numerical methods employed are simply the approximations to the first and second derivatives and the area under a curve. From the potential energy function, the force acting on the particle is calculated and displayed. Together, the graphs of potential energy and force as functions of position strengthen the conceptual relationship between these two quantities. The particle’s position as a function of time is calculated by approximating the second derivative in Newton’s second law. Once the position is known at all times then other quantities such as velocity, acceleration, momentum, kinetic energy, potential energy can be computed. The dynamical quantities are displayed graphically so that the students can compare and contrast the results. For the motion of the classical particle, the total energy of the particle is a constant and the graphs highlight how the changes in kinetic energy and potential energy relate to each other. For a particle launched at a potential hill, the particle is either reflected or transmitted while a particle is always transmitted through a well. This is not the case in quantum mechanics where a particle has some probability of being reflected and transmitted10. The motion of a bound classical particle confined by a truncated parabolic well with negative total energy is oscillatory and the particle can never escape the well. This is in contrast to quantum tunnelling and to the case where there is some probability of a particle being in a region where the classical kinetic energy would be negative. The concept of expectation value can be introduced in a less abstract setting and expectation values can be calculated for a range of dynamical quantities. Also, the concept of uncertainty can be introduced and related to the standard deviation of a set of measurements. The ideas of probability and probability density can be applied to locating the particle and contrasted with the concept of a trajectory for a classical particle. Information on new approaches to teaching the introductory quantum mechanics are discussed on the web site for A New Model Course in Applied Quantum Physics. The site includes a set off instructor resources11. The approach described in this paper for studying the motion of a classical particle can be used as a precursor to the study of a particle as a wave and the applications of wave mechanics. Using this model, students can develop a better background in classical concepts that can help them make more 6 informed comparisons between the predictions of classical physics with those of wave mechanics. APPENDIX The model has a classical particle of mass m whose one dimensional motion is determined by a potential energy function U(x). The force F(x) acting on the particle is determined from Eq. (A1) dU (x ) F (x ) = - . (A1) dx Then Newton’s second law 2 d x (A2) F (x ) = m 2 , dt can be solved numerically to find the position of the particle as a function of time. The basis of the numerical method are the approximations of the first and second derivatives. Consider a single-valued continued function (x) that is evaluated at N equally spaced points x1, x2, …, xN. The first and second derivatives of the function (x) at the point xc where c is an index integer, c = 1, 2, 3, …, N are given by Eqs. (A3) and (A4) respectively. The spacing between adjacent points is x = xc+1 – xc. ж ц d y (x ) y (x c +1) - y (x c - 1 ) c = 2, 3,.... N - 1 з ч = (A3) з d x ч 2Dx и шx = x c жd 2y (x )ц y (x ) - 2y (x ) +y (x ) c = 2, 3,.... N - 1. (A4) з ч = c +1 c c - 1 з 2 ч 2 и d x ш Dx x =x c Eq. (A3) is applied to Eq. (A1) to calculate the force F(xc) and Eq. (A4) is used in Eq. (A2) to compute the position of the particle as a function of time t (Dt )2 x (t ) = F {x (t - Dt )} + 2x (t - Dt ) - x (t - 2Dt ). (A5) m To start the calculation one needs to input the initial conditions for the first two time steps. The particle is restricted to move within a region –L/2 x +L/2 and the potential energy U(x) and hence the force F(x) are known at N positions, x1 = -L/2, … ,xc ,… , xN = +L/2. The force term, F{x(t-t)}, is determined by finding the index c such that x(t-t) ~ xc. This is easy to do since there is a linear relationship between the position, x, and the index c 7 c = integer part {c1 x + c2} (A6) where c1 = (N-1)/L, c2 = 1+ c1 L/2 and then F{x(t-t)} = F(xc). Expectation values and uncertainties can be found for the classical particle. To do this it is necessary to approximate integrals numerically. A simple approximation that is well suited for spreadsheet calculation is N x N y (x )dx = y (x c )Dx . (A7) x 1 е т c =1 Some mathematical packages have commands to evaluate definite integrals. For example, in Matlab the command trapz(X,Y) computes the integral of Y with respect to X using the trapezoidal method. 8 1P.T. Matthews, Introduction to Quantum Mechanics (McGraw-Hill, London, 1968), 2nd ed., p 32. 2D.F Styer, “Common misconceptions regarding quantum mechanics,” Am. J. Phys. 64 (1), 31-34 (1996). 3C. Singh, “Student understanding of quantum mechanics,” Am. J. Phys. 69 (8), 885- 895 (2001). 4W. E. Lamb, “Super classical quantum mechanics: The best interpretation of nonrelativistic quantum mechanics,” Am. J. Phys. 69 (4), 413-422 (2001). 5W.J. Orvis, Excel for Scientists and Engineers (Sybex, San Francisco, 1996), 2nd ed. 6D. Hanselman, B. Littlefield, Mastering Matlab 5 (Prentice-Hall, New Jersey, 1998). 7J. M. Fegan, Quantum Methods with Mathematica (Telos, Springer-Verlag, Sanata Clara, 1994). 8S.T. Thornton and A. Rex, Modern Physics for scientists and Engineers (Saunders College, Fort worth, 2000), 2nd ed., 191-194. 9R. A. Serway, C. J. Moses and C. A. Moyer, Modern Physics (Saunders College, Fort Worth, 1997), 2nd ed., 213-221. 10D. M. Sullivan, Electromagnetic Simulation using the FDTD Method (IEEE Press, New York, 2000), pp136-140. 11 E.F. Redish, R.N. Steinberg, M.C. Wittmann and the Physics Education Research Group at the University of Maryland, A New Model Course in Applied Quantum Physics, http://www.physics.umd.edu/perg/qm/qmcourse/NewModel/index.html 9 Figure captions Fig. 1. The potential energy U(x) functions for a hill with Um = + 100 J and well with Um = - 100 J and the corresponding forces F(x). The half-width of the well is xm = 6.00 m and the track length is L = 20.0 m. The number of data of points used for the calculations is N = 2000. Fig. 2. Time plots showing the motion in terms of x(t), v(t), a(t) and energies K(t), U(t) and E(t) for a particle of mass, m = 10.0 kg launched towards a potential hill with parameters, Um = + 100 J, xm = 6.00 m and with an initial kinetic energy, K = 110 J. Fig. 3. Time plots showing the motion in terms of x(t), v(t), a(t) and energies K(t), U(t) and E(t) for a particle of mass, m = 10.0 kg launched towards a potential hill with parameters, Um = + 100 J, xm = 6.00 m and with an initial kinetic energy, K = 90 J. Fig. 4. Time plots showing the motion in terms of x(t), v(t), a(t) and energies K(t), U(t) and E(t) for a particle of mass, m = 10.0 kg launched towards a potential well with parameters, Um = - 100 J, xm = 6.00 m and with an initial kinetic energy, K = 110 J. Fig. 5. The motion of the particle trapped in a potential well of depth Um = - 100 J and half-width xm = 8.00 m. The initial position of the particle is x(0) = - 6.00 m. The total energy of the particle is constant so that at any time K(t) + U(t) = E(t) = E. Fig. 6. The unscaled probability density of the classical particle oscillating in a potential well. The area under the curve for an interval xp gives the relative probability of locating the particle within this interval. 10 Fig. 1. The potential energy U(x) functions for a hill with Um = + 100 J and well with Um = - 100 J and the corresponding forces F(x). The maximum half-width of the well is xm = 6.00 m and the track length is L = 20.0 m. The number of data of points used for the calculations is N = 2000. 11 Fig. 2. Time plots showing the motion in terms of x(t), v(t), a(t) and energies K(t), U(t) and E(t) for a particle of mass, m = 10.0 kg launched towards a potential hill with parameters, Um = + 100 J, xm = 6.00 m and with an initial kinetic energy, K = 110 J. 12 Fig. 3. Time plots showing the motion in terms of x(t), v(t), a(t) and energies K(t), U(t) and E(t) for a particle of mass, m = 10.0 kg launched towards a potential hill with parameters, Um = + 100 J, xm = 6.00 m and with an initial kinetic energy, K = 90 J. 13 Fig. 4. Time plots showing the motion in terms of x(t), v(t), a(t) and energies K(t), U(t) and E(t) for a particle of mass, m = 10.0 kg launched towards a potential well with parameters, Um = - 100 J, xm = 6.00 m and with an initial kinetic energy, K = 110 J. 14 Fig. 5. The motion of the particle trapped in a potential well of depth Um = - 100 J and half-width xm = 8.00 m. The initial position of the particle is x(0) = - 6.00 m. The total energy of the particle is constant: K(t) + U(t) = E(t) = E. 15 Fig. 6. The unscaled probability density of the classical particle oscillating in a potential well. The area under the curve for an interval xp gives the relative of locating the particle within this interval. 16 of the corresponding observable by generalizing Eqn. (3)