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Total time derivatives of operators in elementary quantum mechanics Mark Andrews Citation: American Journal of Physics 71, 326 (2003); doi: 10.1119/1.1531579 View online: http://dx.doi.org/10.1119/1.1531579 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/71/4?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Madelung representation of damped parametric quantum oscillator and exactly solvable Schrödinger–Burgers equations J. Math. Phys. 51, 122108 (2010); 10.1063/1.3524505 Energy conservation in quantum mechanics Am. J. Phys. 72, 580 (2004); 10.1119/1.1648326 Addendum to: “The one-dimensional harmonic oscillator in the presence of a dipole-like interaction” [Am. J. Phys. 71 (3), 247–249 (2003)] Am. J. Phys. 71, 956 (2003); 10.1119/1.1592514 Fractals and quantum mechanics Chaos 10, 780 (2000); 10.1063/1.1050284 Invariant operators for quadratic Hamiltonians Am. J. Phys. 67, 336 (1999); 10.1119/1.19259 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 Total time derivatives of operators in elementary quantum mechanics Mark Andrews Department of Physics, The Faculties, Australian National University, ACT 0200, Australia 共Received 25 April 2002; accepted 30 October 2002兲 The use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation, including the propagator. © 2003 American Association of Physics Teachers. 关DOI: 10.1119/1.1531579兴 I. INTRODUCTION The expectation value of any operator  changes with time according to1 冓 冔 d i  ⫹ 具 关 Ĥ, 兴 典 . 具  典 ⫽ dt t ប 共1.1兲 The first term on the right-hand side comes from the intrinsic time dependence in Â, and the second from the way the states change with time. It has occasionally been suggested2 that the total time derivative of any operator  be defined as d i Â⬅Â˙ ª Â⫹ 关 Ĥ, 兴 . dt t ប 共1.2兲 冓 冔 共1.3兲 and it can easily be seen that the definition implies that d 共 ÂB̂ 兲 ⫽Â˙ B̂⫹ÂB̂˙ dt 共1.4兲 and d 共 f 共 t 兲  兲 ⫽ ḟ Â⫹ f Â˙ , dt II. FORMAL DEVELOPMENT FOR SYSTEMS DESCRIBED BY A POTENTIAL We first consider Hamiltonians of the form 共The notation XªY indicates that X is defined to be Y.兲 Then it is always true that d d  , 具  典 ⫽ dt dt After showing how the formal development of quantum mechanics and its application to some systems can be simplified, we will find ‘‘invariant operators,’’ that is, operators whose total time derivative is zero. This property is the appropriate generalization of ‘‘constant of the motion’’ to operators that depend on the time. Invariant operators are used here to find general properties and specific examples of the evolution of wave packets and to find the energy eigenvalues of the harmonic oscillator. 共1.5兲 which implies that the total derivative obeys the same algebraic rules as ordinary derivatives. Because this total time derivative is not defined to be a rate of change of anything, some may prefer to use a different symbol for it, for example, D t instead of d/dt, but the latter notation will be used here. The context determines its meaning; when applied to an operator, its meaning is as in Eq. 共1.2兲. Its use eliminates much of the tedious calculation of commutators in conventional treatments and allows calculations that closely follow those of classical mechanics. For many systems it also enables simple methods to be used to obtain the evolution of some wave packets and the propagator. It brings many of the advantages gained by moving to Heisenberg’s picture of the time-evolution while actually remaining in the Schrödinger picture. Ĥ⫽ 1 2 p̂ ⫹V 共 r̂兲 . 2m 共2.1兲 In this case all the operators are independent of the time, so calculating the total time derivative requires only the commutator with the Hamiltonian. A calculation of these commutators gives mr̂˙⫽p̂ and p̂˙⫽⫺ⵜV̂, 共2.2兲 as in classical mechanics. Ehrenfest’s theorem for the time evolution of expectation values follows immediately from Eq. 共1.3兲: m d 具 r̂典 ⫽ 具 p̂典 dt and d 具 p̂典 ⫽⫺ 具 ⵜV̂ 典 . dt 共2.3兲 The virial theorem also follows easily using Eq. 共1.4兲 from the expectation value of 1 d 共 r̂"p̂兲 ⫽r̂˙"p̂⫹r̂"p̂˙⫽ p̂2 ⫺r̂"ⵜV̂. dt m 共2.4兲 The time derivative of the angular momentum is d d L̂⫽ 共 r̂⫻p̂兲 ⫽r̂˙⫻p̂⫹r̂⫻p̂˙⫽⫺r̂⫻ⵜV̂, dt dt 共2.5兲 because p̂⫻p̂⫽0. For a central potential, ⵜV ⫽r ⫺1 (dV/dr)r and therefore L̇⫽0, and L is a constant of the motion. 326 Am. J. Phys. 71 共4兲, April 2003 http://ojps.aip.org/ajp/ © 2003 American Association of Physics Teachers 326 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 These calculations are not only simpler, because the commutators were used only for the equations of motion 共2.2兲, but easier for students because the calculations closely follow the corresponding classical calculations. Next consider how the spread of the position and momentum change with time for a free wave packet in one dimension. There is no force, so mx̂˙ ⫽p̂ and p̂˙ ⫽0. To obtain the spread in position, we use the operator (x̂⫺ 具 x̂ 典 ) 2 . Unlike the above operators, this operator is time dependent. Up to now â˙ has just been a convenient shorthand for iប ⫺1 关 Ĥ,â 兴 , but now the intrinsic time dependence will also play a role. We use the operator equations of motion to find m d 共 x̂⫺ 具 x̂ 典 兲 2 ⫽ 共 x̂⫺ 具 x̂ 典 兲共 p̂⫺ 具 p̂ 典 兲 ⫹ 共 p̂⫺ 具 p̂ 典 兲共 x̂⫺ 具 x̂ 典 兲 , dt 共2.6兲 m2 d2 共 x̂⫺ 具 x̂ 典 兲 2 ⫽2 共 p̂⫺ 具 p̂ 典 兲 2 . dt 2 共2.7兲 If we define (⌬x) 2 ª 具 (x̂⫺ 具 x̂ 典 ) 2 典 and (⌬p) 2 ª 具 (p̂⫺ 具 p̂ 典 ) 2 典 , then ⌬p is constant and m2 d2 共 ⌬x 兲 2 ⫽2 共 ⌬ p 兲 2 . dt 2 and â˙ ⫽0. Invariant operators will be used in this way to produce different solutions of Schrödinger’s equation for several systems. Now consider whether eigenstates of an invariant operator will satisfy Eq. 共3.1兲. The operators 共in Schrödinger’s representation兲 involve differentiation by position only, so if (r,t) is a solution of â ⫽ ␣ , then f (t) (r,t) also will be a solution. Thus it is not true that every eigenfunction will satisfy Schrödinger’s equation. However, if (r,t) is an eigenfunction at a particular time t 0 , then there will be a unique solution (r,t) of Schrödinger’s equation 共for all t兲 such that (r,t 0 )⫽ (r,t 0 ) and, from Eq. 共3.2兲, (â⫺ ␣ ) will satisfy Schrödinger’s equation and therefore will remain zero. That is, (r,t) will be both an eigenfunction of â 共with constant eigenvalue ␣兲 and a solution of Schrödinger’s equation. Thus, if there is only one eigenfunction of â, it must satisfy Eq. 共3.1兲, apart from a time-dependent factor. If there are several linearly independent eigenfunctions of â, then linear combinations with time-dependent coefficients may be required to satisfy Eq. 共3.1兲. Invariant operators will now be found for some simple one-dimensional systems. 共2.8兲 关Note that d 2 具  典 /dt 2 ⫽d(d 具  典 /dt)/dt⫽d( 具 dÂ/dt 典 )/dt ⫽ 具 d 2 Â/dt 2 典 , where d 2 Â/dt 2 means the result of the action in Eq. 共1.2兲 applied twice to Â.] Therefore, ⌬x must have the form ⌬x⫽⌬x 0 冑1⫹ 共 t⫺t 0 兲 2 / 2 , 共2.9兲 where ⫽m⌬x 0 /⌬ p. Hence, for a free wave packet, the spread ⌬x goes through a minimum ⌬x 0 and approaches a linear increase with time at large distances from the minimum. A similar, but slightly lengthier, calculation can be carried out for the harmonic oscillator to show the centroid of every wave packet not only follows the classical oscillation, but its width oscillates at twice the frequency. For a linear potential 共motion under gravity or a uniform electric field兲 the width behaves exactly as for the free case. A comparison of these calculations with the conventional methods3 will show how much simpler this approach is. IV. A FREE PARTICLE The operator equations of motion are mx̂˙ ⫽p̂ and p̂˙ ⫽0. Classically, mx⫺ pt is constant and similarly mx̂⫺p̂t is invariant. A constant can be added to t, even a complex one; but a real constant just changes the origin of time, and we take âªmx̂⫺ p̂ 共 t⫺i 兲 Ĥ ⫽iប â ␣ ªâ⫺ ␣ ⫽m 共 x̂⫺x̄ 兲 ⫺ 共 p̂⫺ p̄ 兲共 t⫺i 兲 . 共3.1兲 it will produce another solution, because 冉 Ĥ⫺iប 冊 冉 冊 â â ⫽ Ĥâ⫺iប ⫺âĤ ⫽⫺iបâ˙ t t 共3.2兲 共4.2兲 Now ␣ will have â ␣ ␣ ⫽0 and therefore 具 â ␣ 典 ⫽0, 具 x̂ 典 ⫽x̄, and 具 p̂ 典 ⫽ p̄ for an eigenstate of â. Furthermore â ␣ ⫺â ␣† ⫽2i 共 p̂⫺ p̄ 兲 , 1 2 , t 共4.1兲 To find an eigenstate ␣ of â with eigenvalue ␣, we first find the real values x̄ 0 , p̄ 0 such that ␣ ⫽mx̄ 0 ⫹ip̄ 0 . Then ␣ ⫽mx̄⫺ p̄(t⫺i ) for all time, where x̄, p̄ are the solutions of the classical equations of motion with initial values x̄ 0 , p̄ 0 . Next define III. INVARIANT OPERATORS An operator is ‘‘invariant’’ if its total time derivative is zero. This definition includes the usual ‘‘constant of the motion,’’ that is, a time-independent operator that commutes with the Hamiltonian. It will be seen below that timedependent invariant operators are especially interesting. It is easy to see that if â and b̂ are invariant then so are â⫹b̂, âb̂, the adjoints â † and b̂ † , and â if is a constant. If an invariant operator â is applied to any solution of Schrödinger’s equation 共 real兲 . 共4.3兲 共 â ␣ ⫹â ␣† 兲 ⫽m 共 x̂⫺x̄ 兲 ⫺ 共 p̂⫺ p̄ 兲 t, and therefore 2m 共 x̂⫺x̄ 兲 ⫽â ␣ 共 1⫺it/ 兲 ⫹â ␣† 共 1⫹it/ 兲 . Hence, using 关 â ␣ ,â ␣† 兴 ⫽ 关 â,â † 兴 ⫽2បm , we have 4 2 具 共 p̂⫺ p̄ 兲 2 典 ⫽ 具 â ␣ â ␣† 典 ⫽ 具 â ␣† â ␣ ⫹2បm 典 ⫽2បm , 4m 具 共 x̂⫺x̄ 兲 2 2 共4.4兲 典 ⫽ 具 â ␣ â ␣† 典 共 1⫹t 2 / 2 兲 ⫽2បm 共 1⫹t 2 / 2 兲 . 共4.5兲 共4.6兲 Thus, for any eigenstate of â, ⌬ p⫽ 冑បm/2 and ⌬x⫽ 冑ប /2m 冑1⫹t 2 / 2 , 共4.7兲 consistent with Eq. 共2.9兲. Note that the minimum value of the uncertainty product ⌬x⌬p is 21 ប; so at its minimum this eigenfunction is a minimum uncertainty wave packet. It is easy to find the form of the wave function ␣ (x,t). From â ␣ ␣ ⫽0, we have 327 Am. J. Phys., Vol. 71, No. 4, April 2003 Mark Andrews 327 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 冋 册 m 共 x⫺x̄ 兲 i , ⫽ p̄⫹ x ប t⫺i and hence ␣ 共 x,t 兲 ⫽exp 共4.8兲 冋 册 i m 共 x⫺x̄ 兲 2 共 t 兲 ⫹p̄x⫹ , ប 2 共 t⫺i 兲 共4.9兲 where (t) can be determined by substituting ␣ (x,t) into Schrödinger’s equation. This substitution gives ˙ ⫽ 21 iប/ 共 t⫺i 兲 ⫺Ē, 共4.10兲 where Ēª 21 p̄ 2 /m, and therefore ␣ 共 x,t 兲 ⫽ 1 ¯ 冑t⫺i e i 共 p̄x⫺E t 兲 /ប exp 冉 冊 im 共 x⫺x̄ 兲 2 . 2ប 共 t⫺i 兲 共4.11兲 Equation 共4.11兲 represents the moving Gaussian wave function found in most texts, but here its expectation values ( 具 x̂ 典 , 具 p̂ 典 ,⌬x,⌬ p) are known without any further integration. The eigenfunction ␣ (x,t), with p̄⫽0 and ⫽0, must 共when appropriately normalized兲 be the propagator for the free particle. The propagator is just the evolution of the ␦ function, and the eigenfunction 共with eigenvalue x̄) of x̂ is ␦ (x⫺x̄). Therefore, the eigenfunction of x̂⫺p̂t/m must approach ␦ (x⫺x̄) as t→0. Thus the propagator K(x,x ⬘ ,t), such that 冕 ⬁ ⫺⬁ K 共 x,x ⬘ ,t 兲 共 x ⬘ ,t 0 兲 dx ⬘ ⫽ 共 x,t 0 ⫹t 兲 V. UNIFORM BUT TIME-VARYING FORCE 冉 冊 im ⌿ 共 x,x ⬘ ,t 兲 ª exp 共 x⫺x ⬘ 兲 . 2បt 冑t 1 共4.13兲 To determine the factor, use the integral ⬁ ⫺⬁ Thus C(y)⫹iS(y) satisfies the free Schrödinger equation if y⫽ 冑m/ បtx, and ª 21 ⫹ 12 (1⫺i) 关 C(y)⫹iS(y) 兴 is the time evolution of the step function and also satisfies âp̂ ⫽0. 关If ªp̂ , then â ⫽0, so that ⫽ f (t) , where is the propagator, and therefore (x,t)⫽(i/ប) f (t) ⫻ 兰 x (x ⬘ ,t)dx ⬘ .] Figure 1 shows 兩 2 兩 . Many other invariants can be made from powers, sums, and products of â and p̂. These can be used to create a vast variety of solutions of the free Schrödinger equation, but they will not be pursued further here. 共4.12兲 for any wave function (x,t), can differ only by a constant factor from 冕 Fig. 1. The free evolution of the step function. The graph shows 兩 2 兩 where (x) is initially the step function. The shape is valid for all times t⬎0; all that changes is the scale on the x-axis. The values of x shown are for t ⫽m/ ប. ⌿ 共 x,x ⬘ ,t 1 兲 ⌿ 共 x ⬘ ,x ⬙ ,t 2 兲 dx ⬘ ⫽ 冑2 iប/m⌿ 共 x,x ⬙ ,t 1 ⫹t 2 兲 , 共4.14兲 which is easily calculated by direct integration. Because the propagator must propagate itself, it must be K(x,x ⬘ ,t) ⫽⌿(x,x ⬘ ,t)/ 冑2 iប/m. Other invariants: For the free particle, p̂ is also invariant 共but independent of t兲. Thus the spatial derivative of any solution of the free Schrödinger equation is also a solution, as are all higher derivatives as follows directly from Schrödinger’s equation. A similar property holds for indefinite spatial integrals and time derivatives and integrals of any solution of the free Schrödinger equation. The spatial derivatives of the propagator are the evolutions of the derivatives of the ␦ function, and can be related4 to Hermite polynomials using 2 2 共 d/d 兲 n e ⫺x ⫽ 共 ⫺1 兲 n H n 共 兲 e ⫺x . 共4.15兲 The integral of the ␦ function is a step function and the integral of the propagator can be expressed in terms of the Fresnel integrals C 共 y 兲 ⫹iS 共 y 兲 ⫽ 冕 y 0 exp共 i z 兲 dz. 1 2 2 共4.16兲 The Hamiltonian is Ĥ⫽ 1 2 p̂ ⫺E共 t 兲 x̂, 2m 共5.1兲 which could represent a charged particle in a time-varying electric field. The operator equations of motion are mx̂˙ ⫽p̂, p̂˙ ⫽E(t) which differ from the free case only in that p̂˙ varies with time, but is still independent of x̂ and p̂. Thus, all we need to do is change the invariant â by adding a suitable time-varying multiple of the unit operator: âªmx̂⫺p̂ 共 t⫺i 兲 ⫹ 共 t 兲 . 共5.2兲 Then â˙ ⫽⫺E(t)(t⫺i )⫹ ˙ , so that 共 t 兲⫽ 冕 t 0 共 t ⬘ ⫺i 兲 E共 t ⬘ 兲 dt ⬘ . 共5.3兲 To generalize this invariant to refer to an arbitrary trajectory, we define â ␣ ªm 共 x̂⫺x̄ 兲 ⫺ 共 p̂⫺ p̄ 兲共 t⫺i 兲 ⫽â⫺ ␣ , 共5.4兲 where x̄, p̄ satisfy the classical equations of motion mx̄˙ ⫽p̄ and p̄˙ ⫽E(t) to ensure that â ␣ is invariant. Here ␣ ⫽mx̄ ⫺ p̄(t⫺i )⫹ (t) and the real and imaginary parts of the constant ␣ give the classical motion p̄⫽ p̄ 0 ⫹ 冕 t 0 E共 t ⬘ 兲 dt ⬘ , mx̄⫽mx̄ 0 ⫹ p̄ 0 t⫹ 冕 t 0 共 t⫺t ⬘ 兲 E共 t ⬘ 兲 dt ⬘ . 共5.5兲 328 Am. J. Phys., Vol. 71, No. 4, April 2003 Mark Andrews 328 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 As before 具 â ␣ 典 ⫽0 gives 具 p̂ 典 ⫽p̄ and 具 x̂ 典 ⫽x̄. Thus 具 x̂ 典 , 具 p̂ 典 exactly follow the classical trajectory. The widths ⌬x, ⌬p are not affected by (t) and therefore evolve exactly as they would if no force were acting. The wave function has the same form as for the free particle 关see Eq. 共4.11兲兴, except that x̄, p̄ now follow a trajectory that depends on the field E(t). 冋 ⫽exp 册 i 共 共 t 兲 ⫹ p̄x 兲 ⫺ 共 x⫺x̄ 兲 2 /L 2 , ប 1 1 2 1 ˙ ⫽⫺ ប ⫺ ¯p ⫹ m 2 x̄ 2 , 2 2m 2 The Hamiltonian is 1 2 1 p̂ ⫹ m 2 x̂ 2 , Ĥ⫽ 2m 2 共6.1兲 and the operator equations of motion are mx̂˙ ⫽ p̂, p̂˙ ⫽⫺m 2 x̂. From these equations, we have x̂˙ ⫹ip̂˙ /m ⫽⫺i (x̂⫹ip̂/m ), and therefore âªe i t 共 x̂⫹ip̂/m 兲 /L 共6.2兲 is invariant. The divisor Lª 冑2ប/(m ) is inserted to make 关 â,â † 兴 ⫽1, as usual. Energy eigenstates: Using this invariant operator allows a slightly different approach to finding the energy eigenstates. When multiplied by exp(⫺iEt/ប), an eigenstate with energy E satisfies Schrödinger’s equation and therefore if â is applied to such an energy eigenstate, it either gives zero or a solution of Schrödinger’s equation with its only time dependence in the factor exp关⫺i(E⫺ប)t/ប兴, that is, an eigenstate with E lower by ប. Similarly â † , which is also invariant, raises the energy by ប. Because the energy eigenvalues must be positive, if we repeatedly apply â to any energy eigenstate, we must eventually reach a ground state 0 such that â 0 ⫽0. Then â † â 0 ⫽0, and Ĥ⫽ប 共 â † â⫹ 21 兲 , 共6.3兲 so the lowest energy eigenvalue is E .0 ⫽ 21 ប , and every eigenvalue must be of the form (n⫹ 21 )ប with n⫽0,1,2,... . The spatial dependence of the wave function 0 (x,t) is found from â 0 ⫽0, which gives 0 共 x,t 兲 ⫽e ⫺it/2 exp共 ⫺x 2 /L 2 兲 . 共6.4兲 Applying â repeatedly to 0 (x,t) gives the excited states, in the usual way. Coherent states: To deal with an eigenstate of â with eigenvalue ␣, we first find the classical motion x̄, p̄ that has real initial values x̄ 0 , p̄ 0 such that ␣ ⫽(x̄ 0 ⫹ip̄ 0 /m )/L. Then e i t (x̄⫹ip̄/m )/L will be constant 共and equal to ␣兲 for all time. Now we define † â ␣ ªâ⫺ ␣ ⫽e i t 关共 x̂⫺x̄ 兲 ⫹i 共 p̂⫺p̄ 兲 /m 兴 /L. 共6.5兲 Then the solutions of â ␣ ⫽0 will have 具 â ␣ 典 ⫽0, and therefore 具 x̂ 典 ⫽x̄ and 具 p̂ 典 ⫽p̄; that is, the wave packet evolves with its centroid at x̄, p̄ following a sinusoidal oscillation. To find the spread in position, we use x̂⫺x̄⫽ 12 L 共 â ␣ e ⫺i t ⫹â ␣† e i t 兲 , 共6.6兲 and obtain 具 共 x̂⫺x̄ 兲 2 典 ⫽ 41 L 2 具 â ␣ â ␣† 典 . 共6.7兲 we get ⌬x⫽ L. This reThen using sult is the same as for the ground state energy eigenstate, 1 2 共6.8兲 and (t) can be found by inserting Eq. 共6.8兲 into Schrödinger’s equation. The result is VI. THE SIMPLE HARMONIC OSCILLATOR 关 â ␣ ,â ␣† 兴 ⫽ 关 â,â † 兴 ⫽1, which corresponds to x̄⫽ p̄⫽0. Similarly, ⌬p⫽ប/L, so these states always have the minimum uncertainty product. The differential equation â ␣ ⫽0 yields 共6.9兲 so that ⫽⫺ 21 ប t⫺ 21 x̄p̄. 共6.10兲 These states, which follow the classical oscillation without change of width, are known as coherent states. The ‘‘raising operator’’ â ␣† can be applied to these states to obtain displaced excited states that are essentially Hermite–Gaussians that also follow a classical oscillation.4 Squeezed states: The operator â in Eq. 共6.2兲 is not the most general invariant that is linear in x̂ and p̂. The form b̂⫽ â⫹ v â † 共6.11兲 will be invariant if and are constants. To set the scale of b̂, we also take 关 b̂,b̂ † 兴 ⫽1, which implies that * ⫺ * ⫽1. In terms of x̂ and p̂, បb̂⫽ 共 t 兲 p̂⫺ 共 t 兲 x̂, 共6.12兲 where ⫽ 12 iL 共 e i t ⫺ e ⫺i t 兲 , ⫽⫺ប 共 e i t ⫹ e ⫺i t 兲 /L. 共6.13兲 Note that and have the dimensions of length and momentum, respectively, and satisfy the classical equations of mo˙ ⫽⫺m 2 , but can be complex. The phase tion, m ˙ ⫽ , assigned to b̂ is not important, so we take to be real. Then adding a constant phase to is equivalent to shifting the time by 21 / 关because a(t)⫹ e i a † (t)⫽ e i t a(0) ⫹ e ⫺i( t⫺ ) a † (0)⫽e 1/2i b̂(t⫺ 21 / )]. Thus, we can also take to be real. Instead of the two parameters and , it is sometimes convenient to use the ‘‘squeezing factor’’ sª ⫹ . Then 2 ⫺ 2 ⫽1 leads to ⫺ ⫽1/s and s⭓1. The coherent states correspond to ⫽1, ⫽0, or s⫽1. Next we introduce បb̂ ␣ ª 共 t 兲共 p̂⫺ p̄ 兲 ⫺ 共 t 兲共 x̂⫺x̄ 兲 , 共6.14兲 where x̄, p̄ is a classical oscillation and again the solution of b̂ ␣ ⫽0 共and therefore any eigenstate of b̂) will follow the classical motion in the sense that 具 x̂ 典 ⫽x̄ and 具 p̂ 典 ⫽p̄. From Eq. 共6.14兲 and using Im(*)⫽ 21ប, we find x̂⫺x̄⫽i 共 * b̂ ␣ ⫺ b̂ ␣† 兲 , p̂⫺p̄⫽i 共 * b̂ ␣ ⫺ b̂ ␣† 兲 , 共6.15兲 and therefore, similarly to the derivation of Eq. 共6.7兲, ⌬x ⫽ 兩 兩 and ⌬p⫽ 兩 兩 . From Eq. 共6.13兲, at t⫽0, ⌬x⫽ 21 L( ⫺ )⫽ 12 L/s, and ⌬ p⫽បs/L. Then 2⌬x/L and L⌬p/ប oscillate between the two extremes of s and 1/s, each taking its maximum value when the other takes its minimum. The differential equation corresponding to b̂ ␣ ⫽0 is easily solved 329 Am. J. Phys., Vol. 71, No. 4, April 2003 Mark Andrews 329 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 关similarly to Eq. 共6.8兲兴 and the squeezed states of the oscillator are also moving Gaussians. The propagator requires ⫽0 to ensure that the eigenfunctions of b̂ will be ␦ (x⫺x̄) and will be zero at t⫽0 if s ⫽⬁. 共A factor of s must be removed from b̂ ␣ to keep it finite in this limit.兲 VII. REPULSIVE INVERSE SQUARE POTENTIAL The Hamiltonian is Ĥ⫽ 1 2 p̂ ⫹c/x 2 . 2m 共7.1兲 Only the repulsive case (c⬎0) will be considered because the attractive case is too singular and requires special treatment.5 Such a potential arises, with c⫽ប 2 ᐉ(ᐉ⫹1)/2m, for the radial part of the wave function 共multiplied by r兲 for a spherically symmetric system in three dimensions, and the spherical Bessel functions are familiar as the radial part of the energy eigenfunctions for the free particle. We will treat the general case of arbitrary c⬎0 in one dimension. The repulsive hill is impenetrable so there can be no tunneling through the origin. Therefore, only x⬎0 need be considered, and we expect the wave function to vanish as x→0. The operator equations of motion are mx̂˙ ⫽p̂, ˙p̂ ⫽2c/x 3 . Therefore d 1 d 共 x̂p̂ 兲 ⫽ 共 p̂x̂ 兲 ⫽ p̂ 2 ⫹2c/x 2 ⫽2Ĥ dt dt m 共7.2兲 ⪠21 共 x̂p̂⫹p̂x̂ 兲 ⫺2tĤ 共7.3兲 and is invariant. 共We have used the symmetrized form to keep it Hermitean.兲 Furthermore, 1 1 d 2 m 共 x̂ 兲 ⫽ 共 x̂ p̂⫹p̂x̂ 兲 ⫽â⫹2tĤ, 2 dt 2 共7.4兲 so that b̂ª 21 mx̂ 2 ⫺tâ⫺t 2 Ĥ⫽ 21 mx̂ 2 ⫺ 21 共 x̂ p̂⫹p̂x̂ 兲 ⫹t 2 Ĥ 共7.5兲 is also invariant. It follows that 具 x 典 is quadratic in t for any state. Also the eigenfunction of b̂, with arbitrary eigenvalue , will give the propagator for the system because as t→0, it must approach an eigenfunction of x̂ 2 , that is, ␦ (x 2 ⫺2  /m) apart from a factor. 共The variable x 2 is just as good as x for this system because we need only consider positive values of x.兲 The solution of the differential equation b̂ ⫽  is 2 ⫽exp共 i 21 mx 2 /បt 兲 冑xJ 共 冑2m  /ប 2 x/t 兲 , 共7.6兲 where ª ⫹2cm/ប . A discussion of how to find this solution is given in the Appendix. The positive value of gives the regular solution; the negative value also gives a solution, but it is singular at x⫽0. As before, this eigenfunction requires a time-dependent factor to also satisfy Schrödinger’s equation. The factor can be found to be exp(i/បt)/t by inserting in Eq. 共7.6兲 into Schrödinger’s equation. Thus 2 1 4 2 ⫽exp关 i 共  ⫹ 12 mx 2 兲 /បt 兴共 冑x/t 兲 J 共 冑2m  /ប 2 x/t 兲 共7.7兲 is the eigenfunction of b̂ that also evolves as a wave function. As before, a more general wave function will be obtained by replacing t by t⫺i . The propagator results from the limit →0, while for sufficiently large , the wave function corresponds to a localized ‘‘particle’’ coming toward the origin, being slowed to a standstill by the repulsive force and then retreating, as in the classical case. The details of this calculation, and the exact normalization and calculation of 具 x 2 典 , 具 Ĥ 典 , and 具 â 典 , will be pursued in the Appendix. VIII. RELATION TO THE HEISENBERG PICTURE For the operator Â, the equivalent in the Heisenberg picture6 is  H ªT̂ † ÂT̂, where T̂ is the time-evolution operator such that (t)⫽T̂ (0) and iបdT̂/dt⫽ĤT̂. It follows that ih  d H ⫽ 关  H ,Ĥ H 兴 ⫹ihT̂ † T̂. dt t 共8.1兲 That is, d H † d ⫽T̂ T̂ . dt dt 共8.2兲 共Note that where d/dt is applied to an operator in Heisenberg’s picture, it does not have the special meaning it has when applied to an operator in Schrödinger’s picture兲. Thus the total time derivative of any operator is the Schrödinger picture version of the time derivative of its Heisenberg picture version, which implies that the operator equations of motion will have the same form in the Heisenberg picture as they have in Schrödinger picture using the total time derivative. The formal development in Sec. II and the determination and application of the invariant operators could therefore be presented in the Heisenberg picture, but the advantage of using the total time derivative in an elementary course is that the Heisenberg picture requires much more mathematical machinery. One must deal with exponentials of operators and, for Hamiltonians that depend on the time, the determination of the explicit form of the transformation is often quite difficult. IX. DISCUSSION Time-dependent invariant operators that are linear in x̂ and p̂ can be found4 for any Hamiltonian that is quadratic 共or linear兲 in x̂ and p̂, even with arbitrary time dependence in the Hamiltonian. These systems include, for example, harmonic oscillators that are driven 共a time-varying linear term in H兲 or have varying frequency, as well as the cases treated previously. The case of V⫽c/x 2 shows that the method is not restricted to quadratic Hamiltonians. However, I have not found a useful invariant for the important case of V⫽c/x. An indication that this potential might be more difficult can be seen from the corresponding classical system. For all classical one-dimensional potentials, an integral gives the time as a function of x and the energy. In the previous examples, the time 共or some function of the time兲 can be expressed in a form that is linear in the velocity and the energy and then the eigenfunctions of the corresponding quantum operator can be found easily. But for V⫽c/x, the classical result is highly non-linear in the energy. 330 Am. J. Phys., Vol. 71, No. 4, April 2003 Mark Andrews 330 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 In summary, the use of the total derivative of operators 共in Schrödinger’s picture of the time evolution兲 simplifies the formal development of quantum mechanics and its application to particular systems. This approach eliminates many of the tedious calculations of commutators that are required in the usual treatment and allows a development that more closely follows the familiar classical treatment. Furthermore, it makes it easy to use the simple invariant operators that can be found for many of the systems often studied in elementary quantum mechanics to find simple wave packets that illustrate the dynamics of the system and to find the propagator. APPENDIX: ANALYSIS OF THE WAVE FUNCTION FOR THE POTENTIAL VÄcÕx 2 The equation b̂ ⫽  is, in Schrödinger’s representation, an ordinary differential equation 共ODE兲 in x with t as a parameter. Its solution, apart from a possible time-dependent factor, will satisfy Schrödinger’s equation, which is a partial differential equation 共PDE兲. But b̂ ⫽  is not easy to solve even though it is only an ODE. 关The solution was given in Eq. 共7.7兲.兴 Equation-solving programs such as Maple or Mathematica will probably not solve it because they will not attempt to factor out the required factor of 冑x exp( 12imx2/បt) to reduce the equation to Bessel’s form. Surprisingly, one way to obtain the solution is to convert the equation to a PDE using the requirement that the solution also satisfy Schrödinger’s equation. First write b̂ª 21 mx̂ 2 ⫺t(x̂p̂⫺ 21 iប)⫹t 2 Ĥ, and then insert p̂⫽⫺iប / x and Ĥ ⫽iប / t to give the PDE, tx 冋 冉 2 1 1 i  ⫺ mx 2 ⫹t ⫹ t⫹ x t 2 ប 2 冊册 ⫽0. 共A1兲 Such first-order, linear PDEs can be solved easily by standard methods 共or using equation-solving programs兲, and it is easy to verify directly that ⫽exp关i(⫹ 21mx2)/(បt)兴 f(x/t)/冑x satisfies Eq. 共A1兲, if f is an arbitrary function of x/t. If we substitute this form of into Schrödinger’s equation, we see that f (u), with u⫽x/t, must satisfy the ODE u 2 f ⬙ ⫺u f ⬘ ⫹ 共 c 1 ⫹c 2 u 2 兲 f ⫽0, 共A2兲 where c 1 ⫽ 34 ⫺2cm/ប 2 and c 2 ⫽2m  /ប 2 . It is not difficult to recognize that Eq. 共A2兲 can be converted to Bessel’s equation by dividing f (u) by u. Thus ⫽exp关 i 共  ⫹ mx 兲 / 共 បt 兲兴 uJ 共 冑2m  /ប u 兲 / 冑x, 2 1 2 2 共A3兲 where ⫽1⫺c 1 ⫽ ⫹2cm/ប . The form of Eq. 共A3兲 agrees with Eq. 共7.7兲. Expectation values: As before, t can be complex and the function 2 1 4 冋 册 1 i  ⫹ 2 mx 2 冑x ⫽exp J ប t⫺i t⫺i 冉冑 2m  x ប 2 t⫺i 冊 B̂ª mx̂ ⫺ 共 t⫺i 兲共 x̂ p̂⫹ p̂x̂ 兲 ⫹ 共 t⫺i 兲 Ĥ, 2 1 2 ⫻ 共A4兲 2 共A5兲 and satisfies the time-dependent Schrödinger equation for the potential V⫽c/x 2 . To calculate expectation values, we first normalize the wave function using 1 2 共  r ⫹ 2 mx 2 兲 ⫹  i t ប t 2⫹ 2 册 x J 共 a * x 兲 J 共 ax 兲 , t ⫹2 共A6兲 2 where aª 冑2m  /ប 2 /(t⫺i ) and  ⫽  r ⫹i  i . Now apply the integral 冕 ⬁ 0 exp共 ⫺ 21 x 2 兲 J 共 ax 兲 J 共 bx 兲 x dx ⫽ ⫺1 exp关 ⫺ 21 ⫺1 共 a 2 ⫹b 2 兲兴 I 共 ab/ 兲 共A7兲 关for Re()⬎⫺1] given by Weber in 1868. We substitute b ⫽a * and ⫽2 m/ប(t 2 ⫹ 2 ), and obtain 7 冕 * ⬁ dx⫽ 0 冉 冊冉 冊 r 兩兩 ប exp ⫺ I . 2m ប ប 共A8兲 To determine 具 x̂ 2 典 we differentiate Weber’s integral in Eq. 共A7兲 with respect to to obtain8 冕 ⬁ 0 exp共 ⫺ 21 x 2 兲 J 共 ax 兲 J 共 bx 兲 x 3 dx ⫽2 ⫺2 e ⫺w 兵 共 ⫹1⫺w 兲 I 共 z 兲 ⫹zI ⫹1 共 z 兲 其 , 共A9兲 where wª (a ⫹b )/ and zªab/. Equation 共A9兲 implies that 2 1 2 2 冋 册 ប t I ⫹1 共 z 兲  r 1 ⫹1⫹z ⫺ , m 具 x̂ 2 典 ⫽  r ⫹  i ⫹ 共 t 2 ⫹ 2 兲 2 2 I 共 z 兲 ប 共A10兲 and z⫽ 兩  兩 /ប . But Eq. 共A5兲 can be written as B̂ª 21 mx̂ 2 ⫺ 共 t⫺i 兲 â⫺ 共 t 2 ⫹ 2 兲 Ĥ, 共A11兲 and therefore, taking the real and imaginary parts of the expectation value of B̂,  i ⫽ 具 â 典 , and 1 2 m 具 x̂ 2 典 ⫽  r ⫹ 共 t/ 兲  i ⫹ 共 t 2 ⫹ 2 兲 具 Ĥ 典 . 共A12兲 If we compare this form with Eq. 共A10兲, we see that 具 Ĥ 典 ⫽ 冋 册 ប I ⫹1 共 z 兲  r ⫹1⫹z ⫺ . 2 I 共 z 兲 ប 共A13兲 The propagator: A more symmetrical form for the eigenfunction of b̂, with eigenvalue 21 mx ⬘ 2 , is 2 is the eigenfunction, with eigenvalue , of the invariant operator 1 2 冋 * ⫽exp ⫺ ⌿ 共 x,x ⬘ ,t 兲 ⫽ 冑xx ⬘ t 冋 exp 册冉 冊 im 2 mxx ⬘ . 共A14兲 共 x ⫹x ⬘ 2 兲 J 2បt បt As shown in Sec. V, this eigenfunction must be the propagator apart from a factor independent of x and t that will now be determined. The eigenfunction satisfies Schrödinger’s equation and therefore should propagate itself. If we use the integral in Eq. 共A7兲, we can evaluate the relevant integral to give 冕 ⬁ 0 ⌿ 共 x,x ⬘ ,t 1 兲 ⌿ 共 x ⬘ ,x ⬙ ,t 2 兲 dx ⬘ ⫽i ⫹1 共 ប/m 兲 ⌿ 共 x,x ⬙ ,t 1 ⫹t 2 兲 , 共A15兲 where the relation I (iz)⫽i J (z) 关for Im(z)⬎0] has been used. Hence the propagator K, such that 331 Am. J. Phys., Vol. 71, No. 4, April 2003 Mark Andrews 331 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43 冕 ⬁ 0 K 共 x,x ⬘ ,t 兲 共 x ⬘ ,t 0 兲 dx ⬘ ⫽ 共 x,t 0 ⫹t 兲 共A16兲 for any wave function (x,t), is K 共 x,x ⬘ ,t 兲 ⫽i ⫺ 共 ⫹1 兲 共 m/ប 兲 ⌿ 共 x,x ⬘ ,t 兲 . 共A17兲 The single-hump regime: The argument uªax of the Bessel function in has 兩 u 2 兩 ⫽zx 2 . But * contains a factor exp(⫺ 12x2). Hence if 兩  兩 Ⰶប , that is zⰆ1, then 兩兩 will become small 共because of the exponential factor兲 before the argument of the Bessel function leaves the region where 兩 u 兩 Ⰶ1, and therefore8 J (u)⬇( 21 u) /⌫( ⫹1). In this regime the spatial form of is x ⫹1/2 exp关 21imប⫺1x2/(t⫺i)兴, and 兩兩 has just a single hump. If we substitute I (z)⬇( 21 z) /⌫( ⫹1) into Eq. 共A13兲, we obtain 具 Ĥ 典 ⬇ 21 ប ⫺1 ( ⫹1), and 1 2 m 具 x̂ 2 典 ⬇ 具 Ĥ 典 共 t 2 ⫹ 2 兲 . 共A18兲 Equation 共A18兲 can be compared with the classical motion 1 2 mx 2 ⫽Et 2 ⫹ 21 mc/E. 共A19兲 The first term Et 2 corresponds as closely as possible to the quantum term 具 Ĥ 典 t 2 , while the second term, specifying the closest approach to the origin, is 具 Ĥ 典 2 ⬇ 41 ប 2 ( ⫹1) 2 / 具 Ĥ 典 in the quantum case. This quantity is to be compared with 1 1 1 2 1 2 2 mc/E noting that 2 mc⫽ 4 ប ( ⫺ 4 ). The closest approach in the quantum case has a lower limit for 21 m 具 x̂ 2 典 of 9 2 16 ប / 具 Ĥ 典 for a very weak potential (c⬇0) and is always greater than in the classical case with the same energy. This behavior is due to the existence of a completely repulsive barrier at x⫽0 共because is always zero there兲, and therefore is squeezed against this barrier, whereas the classical particle will approach the origin arbitrarily closely for sufficiently small c. In this regime, the wave packet motion is similar to that of the classical particle coming toward the origin, slowing to a standstill, and then retracing its inward trajectory; however, there are quantum effects that are greatest near the closest approach to the origin. Figure 2 shows two examples of the evolution of this wave function; one has zⰆ1 and therefore has a single hump while the other has zⰇ1 and is shown at a time when it has two maxima. Fig. 2. Two wave packets for the potential V⫽c/x 2 . In each case the wave function is as in Eq. 共A4兲; 兩 2 兩 is plotted with m⫽ប⫽c⫽1 and eigenvalue 1  ⫽ 2 . In case 共a兲, ⫽10 and the wave function is shown for t⫽0 共solid兲 and for t⫽20 共dashed兲. This case is in the ‘‘single-hump’’ regime. In case 共b兲, ⫽0.05 and the wave function is shown for t⫽0 共solid line兲 and for t ⫽2 共dashed line兲. For t⫽0, the maximum value is about 2.5. This case is in the regime where the Bessel function gives rise to multiple maxima. Eugen Merzbacher, Quantum Mechanics, 2nd ed. 共Wiley, New York, 1970兲, p. 165. 2 L. D. Landau and E. M. Lifshitz, Quantum Mechanics 共Pergamon, New York, 1958兲, Chap. 9. Also see Ref. 1, p. 337, Eq. 共15.13兲. 3 Eugen Merzbacher, Quantum Mechanics, 3rd ed. 共Wiley, New York, 1998兲, p. 49, Problems 2 and 3. 4 Mark Andrews, ‘‘Invariant operators for quadratic Hamiltonians,’’ Am. J. Phys. 67, 336 –343 共1999兲. 5 For a recent discussion of the attractive case 共in three dimensions兲, see Sydney A. Coon and Barry R. Holstein, ‘‘Anomalies in quantum mechanics: The 1/r 2 potential,’’ Am. J. Phys. 70, 513–519 共2002兲. 6 See Ref. 3, Chap. 14, Sec. 2. 7 G. N. Watson, Theory of Bessel Functions, 2nd ed. 共Cambridge University Press, Cambridge, 1944兲, Chap XIII, Sec. 13.31, p. 395. 8 For relations involving Bessel’s functions, see Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions 共NBS, Washington, DC, 1964兲, Chap. 9. 1 332 Am. J. Phys., Vol. 71, No. 4, April 2003 Mark Andrews 332 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.203.176.45 On: Mon, 01 Sep 2014 05:04:43