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Transcript

QM 480 “On the Shoulders of Giants” An Introduction to Classical Mechanics QM 480 If I have seen further it is by standing on the shoulders of giants. Isaac Newton, Letter to Robert Hooke, February 5, 1675 English mathematician & physicist (1642 1727) QM 480 Quantum Mechanics (QM) is based on classical mechanics. It combines classical mechanics with statistics and statistical mechanics. For native English-speakers, it is somewhat unfortunate that it uses the word “quantum”. A better English word which describes the thrust of this approach would be “pixel”. QM 480 2nd Century BC Lights! Camera! Action! Hero of Alexandria found that light, traveling from one point to another by a reflection from a plane mirror, always takes the shortest possible path. 1657 Pierre de Fermat reformulates the principle by postulating that the light travels in a path that takes the least time! In hindsight, if c is constant then Hero and Fermat are in complete agreement. Based on his reasoning, he is able to deduce both the law of reflection and Snell’s law (nsinQ = n’ sinQ’) QM 480 An Aside Fermat is most famous for his last theorem: Xn +Yn = Zn where n=2 and … On his deathbed, he wrote: And n= arrgh! I’m having a heartattack! His last theorem was only solved by computer in the last 10 years… QM 480 1686 Now we wait for the Math The calculus of variations is begun by Isaac Newton 1696 Johann and Jakob Bernoulli extend Newton’s ideas QM 480 1747 Pierre-Louise-Moreau de Maupertuis asserts a “Principle of Least Action” More Theological than Scientific “Action is minimized through the Wisdom of God” His idea of action is also kind of vague Now we can get back Action (today’s definition)— Has dimensions of length x momentum or energy x time Hmm… p * x or E*t … seems familiar… QM 480 To the Physics 1760 Joseph Lagrange reformulates the principle of least action The Lagrangian, L, is defined as L=T-V where T= kinetic energy of a system and V=potential energy of a system QM 480 Hamilton’s Principle 1834-1835 William Rowan Hamilton’s publishes two papers on which it is possible to base all of mechanics and most of classical physics. Hamilton’s Principle is that a particle follows a path that minimizes L over a specific time interval (and consistent with any constraints). A constraint, for example, may be that the particle is moving along a surface. QM 480 Lagrange’s Equations Recall dU(x) F mx dx Rearrangin g dU d(-U) mx 0 and mx 0 dx dx d d mx 2 mx mx dt dt x 2 so d(-U) d mx 2 0 dx dt x 2 QM 480 Lagrange’s Equations Now mx 2 0 and x 2 U ( x) 0 x And I can add zero to anything and not change the result d mx 2 d mx 2 -U ( x) U ( x) 0 dx 2 dt x 2 but mx 2 T 2 Thus and dL d L 0 dx dt x L T V QM 480 Expanding to 3 Dimensions Since x, y, and z are orthogonal and linearly independent, I can write a Lagrange’s EOM for each. In order to conserve space, I call x, y, and z to be dimensions 1, 2, and 3. So dL d L 0 i 1,2,3 dqi dt qi Amusingly enough, 1, 2, 3, could represent r, q, f (spherical coordinates) or r, q, z (cylindrical) or any other 3-dimensional coordinate system. QM 480 Example: Simple Harmonic Oscillator Recall for SHO: V(x)= ½ kx2 and let T=1/2 mv2 Hooke’s Law: F=-kx dL d L 0 dq dt q 1 2 1 2 L mx kx 2 2 dL d 1 2 kx kx dx dx 2 L 1 2 d mx mx and mx mx x x 2 dt so kx mx 0 or kx mx QM 480 Tip The trick in the Lagrangian Formalism of mechanics is not the math but the proper choice of coordinate system. The strength of this approach is that 1. Energy is a scalar and so is the Lagrangian 2. The Lagrangian is invariant with respect to coordinate transformations Two Conditions Required for QM 480 Lagrange’s Equations 1. The forces acting on the system (apart from the forces of constraint) must be derivable from a potential i.e. F=-dU/dx or some similar type of function 2. The equations of constraint must be relations that connect the coordinates of the particles and may be functions of time. QM 480 Your Turn Projectile: Go to the board and work a simple projectile problem in cartesian coordinates. Don’t worry about initial conditions yet. Now do the same in polar coordinates. Hint: 2 1 2 1 mr m rq 2 2 U mgr sin q T QM 480 Introducing the Hamiltonian First, any Lagrangian which describes a uniform force field is independent of time i.e. dL/dt=0. L L(q, q ) dL L q L q L dt q t q t t dL L L q q dt q q L d L Since q dt q dL d L L q q dt dt q q QM 480 Introducing the Hamiltonian dL d L L d L q q q dt dt q q dt q d L dL d L 0 q q L dt q dt dt q So L q L a constant H q Hmmm… H for Hornblower or Hamilton? QM 480 Introducing the Hamiltonian L L H q L T (q ) V (q ) T q q q 1 T T If T mq 2 mq q mq 2 2T 2 q q So q 2T (T V ) T V H but T V E (mechanical energy!) QM 480 H is only E when It is important to note that H is equal to E only if the following conditions are met: The kinetic energy must be a homogeneous quadratic function of velocity The potential energy must be velocity independent While it is important to note that there is an association of H with E, it is equally important to note that these two are not necessarily the same value or even the same type of quantity! Making Simple Problems QM 480 Difficult with the Hamiltonian Most students find that the Lagrangian formalism is much easier than the Hamiltonian formalism So why bother? Making Simple Problems QM 480 Difficult with the Hamiltonian First, we need to define one more quantity: generalized momenta, pj L pj where j 1,2,3 q j So H q L L becomes q 3 H pq L or H p j q j L j 1 QM 480 SHO with the Hamiltonian 1 2 1 2 L mx kx 2 2 L p p2 p mx x 2 x 2 q m m So H pq L becomes p p2 p2 1 2 H p L kx m m 2m 2 p2 1 2 H kx 2m 2 Big deal, right? But look what we did L=f(q,dq/dt,t) H=f(q,p,t) So our mechanics all depend on momentum but not velocity Recall light has constant velocity, c, but a momentum which is p=hc/l ! QM 480 The Big Deal So if we are going to define mechanics for light, it does not make any sense to use the Lagrangian formulation, only the Hamiltonian! QM 480 That Feynman Guy! Richard Feynman thought that Lagrangian mechanics was too powerful a tool to ignore. Feynman developed the path integral formalism of quantum mechanics which is equivalent to the picture of Schroedinger and Dirac. So which is better? Both and Neither There seems to be no undergraduate treatment of path integral formalism. QM 480 Hamilton’s Equations of Motion Just like Lagrangian formalism, the Hamiltonian formalism has equations of motion. There are two equations for every degree of freedom They are H q p H p q QM 480 p2 1 2 H kx 2m 2 H p x p m H p kx or q dp Since F p dt F kx Finishing the SHO p kx Hooke’s Law again! QM 480 H q p H p q Symmetry Note that Hamilton’s EOM are symmetric in appearance i.e. that q and p can almost be interchanged! Because of this symmetry, q and p are said to be conjugate QM 480 Definition of Cyclic Consider a Hamiltonian of a free particle i.e. H=f(p)… then – dp/dt=0 i.e. momentum is a “constant of the motion” Now in the projectile problem, U=-mgy and for x-component, H=f(px) only! Thus, px= constant and the horizontal variable, x is said to “cyclic”! A more practical definition of cyclic is “ignorable” and modern texts sometimes use this term. QM 480 Definition of canonical Canonical is used to describe a simple, general set of something … such as equations or variables. It was first introduced by Jacobi and rapidly gained common usuage but the reason for its introduction remained obscured even to contemporaries Lord Kelvin was quoted as saying “Why it has been so called would be hard to say” QM 480 Poisson Brackets Poisson Bracket of u and v with respect to the canonical variables q and p is defined as u v v u {u , v} q p q p What if u and v were functions of q and p? {qi , q j } 0 Example {x, y} x y y x 0 x p x y p y { pi , p j } 0 {qi , p j } i , j { pi , q j } QM 480 Kronecker Delta i,k=1 if i=k i,k=0 if i≠k QM 480 Back to Fish Consider two continuous functions g(q,p) and h(q,p) If {g,h}=0 then h and g are said to commute In other words, the order of operations does not matter If {g,h}=1 then quantities are canonically conjugate • A look ahead: we will find that canonically conjugate quantities obey the Uncertainty principle QM 480 Properties of Fish The following are properties of the Poisson Bracket dg g a) {g , H } dt t where H Hamiltonia n H H(q, p) b) q j {q j , H } c) p j { p j , H } QM 480 Levi-Civita Notation Consider t he vector product of A and B C AB The individual components can be expressed in a compact notation Ci ijk A j Bk j ,k where ijk if any index equals any other 0 1 if i, j, k is an even permuation of 1, 2,3 - 1 if i, j, k is an odd permutatio n (out of order) 122 112 133 0 123 312 231 1 213 321 132 1 QM 480 Levi-Civita Notation Consider t he vector product of A and B 1̂ 2̂ C A1 A2 B1 B2 Consider 3̂ A3 B3 C1 A2 B3 A3 B2 123 1 132 1 C1 123 A2 B3 132 A3 B2