Quantum Mechanics: Arrival to Schrodinger Equation

An introduction to quantum mechanics: arrival to Schrodinger equation Shyamal Biswas1 School of Physics, University of Hyderabad, C.R. Rao Road, Gachibowli, Hyderabad-500046, India Date: July 6, 2014 Abstract Introduction to Schrodinger equation is rarely found in standard text books on quantum mechanics which is very often introduced from Planck’s theory on black body radiation problem. Planck’s theory although is good enough to explain the black body radiation problem, which actually is a many body problem of photons in thermal equilibrium, can actually be explained from Planck-Einstein formula for energy of a photon2 and Bose statistics for energy distribution of photons. Essentially a quantum field theory is rooted in to the black body radiation problem. So, indirectly, quantum mechanics is very often introduced not for a purely mechanical system rather for a thermodynamic system of large number of photons. In this note (teaching material), we tried to introduce quantum mechanics, for problems of a purely mechanical system, in particular, from Hamilton’s optical- mechanical analogy and the facts of wave-particle duality. Adequacy and inadequacy of classical mechanics Let us start from Newton’s equation of motion3: dp ✞ ☎ = f~(r) V(r). 1 dt ≡ −∇ ✝ ✆ 1For any correspondence, please email to <sbsp [at] uohyd.ac.in>. 2Planck-Einstein formula for energy of a photon was given in the early 20th century. That time photon was known as light’s energy quanta. 3Sir Isaac Newton gave the equation in around 1687. 1 It is the equation of motion for a point particle whose position, with respect to a locally inertial frame (OXYZ), is given by r = xî + yˆj + zkˆ = r rˆ = rrˆ, and momentum with | | respect to the same inertial fame is given by p = pxî + py ˆj + pzkˆ = p pˆ = ppˆ. Here f~(r) is the force acting on the particle, and for conservative case, it is equal| | to V(r) where V(r) is called as potential energy. Let us clarify something. OXYZ is−∇ an orthogonal (Cartesian) coordinate frame. Any coordinate fame by default is locally inertial. Let us ask a few question whose answers are yet to be known for clarity. What is the meaning of locally inertial frame? For a certain observation in a laboratory with respect to a laboratory fixed frame, if the law of force ( f~(r)) acting on a particle is unaltered for all possible motion in the laboratory, then the lab fixed frame of reference can be called as locally inertial frame. What is meant by motion? Motion of a particle at time t is defined as its mechanical state denoted by r(t), p(t) or simply by r, p 4. What is equation of motion? Implicit relation between{ r(t)} and p(t) in terms{ of} force is called equation of motion. Why is the equation of motion necessary at all? It is necessary to know the time evolution of the mechanical state r(t), p(t) if the initial mechanical state r(0), p(0) is known. Newton gave an equation{ of motion} as in Eqn.(1). It is certainly{ a differential} equation. But, what kind of differential equation it is? In Newtonian formalism (like Lagrangian or Hamiltonian formalism), momentum dr by definition is independentof position, as because, it is defined as p = m dt = mv where dr m is the mass of the particle and v = dt is the velocity of particle. From the definition of momentum, the implicit relation between position and momentum i.e. Newton’s equation of motion, can be recast into a second order differential equation for position: d2r ✞ ☎ m = f~(r). 2 dt2 ✝ ✆ As an example, let us consider the case of a 1-D harmonic oscillator of angular frequency ω. For this case, Newton’s equation of motion takes the form d2 x ✞ ☎ = ω2 x. 3 dt2 − ✝ ✆ We can easily solve this (second order) differential equation, and write the general so- 4It was a brilliant thought of Newton that only two variables r, r˙ are enough to determine the mechanical ... state r, r˙, r¨, r ,... of a system once a correct equation of state are{ given.} { } 2 lution as p (0) ✞ ☎ x(t) = x(0) sin(ωt) + x cos(ωt). 4 mω ✝ ✆ dx(t) There from we can easily get the time evolution of momentum using px(t) = m dt as ✞ ☎ p (t) = p (0) sin(ωt) + mx(0)ω cos(ωt). 5 x − x ✝ ✆ Although position and momentum in Newtonian formalism are independent, they are implicitly related by the equation of motion. Solving the equation of motion (second order differential equation) we can in principle know the time evolution of position (r(t)) and there from that of momentum (p(t)) using its definition. Important thing is that, we can uniquely know simultaneous set of r(t), p(t) . But, to know the same we need to know the initial state r(0), p(0) . This ends{ our little} journey to the Newtonian formalism in classical mechanics.{ To come} to the inadequacy of classical mechanics, let us make a little journey to the other formalisms of classical mechanics. Little journey to Lagrangian and Hamiltonian formalism In Lagrangian formalism5, coordinates and their corresponding velocities are considered to be independent, and the equations of motion are obtained from a general princi- 6 t2 ple called least action principle . Accordingto this principle, action S = L(q1, q2, q3, t1 ...qs, q˙1, q˙2, q˙3,..., q˙ s, t)dt of a mechanical system (eg. a point particle,R a system of point particles, etc) is to be extremized. Here q1, q2, q3,...qs are the minimum number (s 3N) of coordinates necessary to specify uniquely the position of the system, dq≤j q˙ j = dt is the velocity of the system correspondingto the coordinate q j, and L(q, q˙, t) is suitable function (called Lagrangian) of the generalized coordinates ( q ), generalized { j} 5It was originally introduced without the principle of least action (but within calculus of variations) by Leonhard Euler in around 1744 and by Joseph-Louis Lagrange in around 1788. 6The known form of least action principle was introduced by William Rowan Hamilton in around 1835. Before this a different form of least action (time) principle was introduced for optics by Pierre de Fermat in around 1662. Fermat’s principle was later modernized to an action principle by Gottfried Leibniz, Johann Bernoulli and Pierre Louis Maupertuis defining the action as an integral of its inverse speed along its path length in around 1707-1746. Leonhard Euler (and, possibly, Leibniz) defined action for a material particle as the integral of the particle’s speed along its path through space in around 1744. 3 velocities ( q˙ ) and time (t), such that correct equation of motion can be reproduced { j} extremizing the action. For fixed values of q(t1) and q(t2), q may have infinite possi- bilities for each intermediate values of t. But for only one solution q(t), the action is minimized. This solution q(t) is the correct path. Variations of paths around the correct path leads to the Lagrange’s equation of motion7 d ∂L ∂L ✞ ☎ = . 6 dt ∂q˙ j ∂q j ✝ ✆ ∂L In this formalism, generalized momentum is defined as p j = . Now, all together, ∂q˙ j we actually have s second order differential equations. The Lagrange’s equation of motion, to a certain extent, is a generalization of Newton’s equation of motion. Newton’s equation is there inside the correct form of the Lagrangian and subsequent Lagrange’s equation of motion. But, the least action principle is actually a generalized principle which accommodates Newtonian formalism at least for the conservative cases. La- grangian formalism is not only restricted for the description of motion of a mechanical system but also for fields, etc. It principley does not work for dissipative cases. But, Newtonian formalism works for this case too. Thus, for a mechanical system, once Lagrange’s equation(s) of motion is/are given, there is principley no difference between the two formalisms. There of course may be some advantage and disadvantage of the two formalisms in evaluating time evolution of the mechanical state for complex type of motion of the system. In Lagrangian formalism, if the Lagrangian has no explicit time dependence, E = s q˙ p L(q, q˙, t) can be interpreted as an integral of motion (constant), and can j=1 j j − Pbe defined as the energy of the system. In Hamiltonian formalism, coordinates and their corresponding momenta are considered to be independent, and an Hamiltonian is defined as H(q, p, t) = s q˙ p L(q, q˙, t). This definition and Lagrange’s equation j=1 j j − P 7Lagrange originally obtained the equation of motion directly from the Newtonian formulation of mechanics introducing Lagrange multiplier in around 1788. Like Lagrange’s equation of motion, d’Alembert’s principle ( (fi mirï) δri = 0) introduced in around 1744 is considered to be another alternative of New- i − · ton’s equationP of motion particularly for constrained motion. D’Alembert’s principle, however, was possibly written previously in the variational form by Lagrange. D’Alembert’s contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces N ∂ri (Q j = fi ) do not need constraint forces to be included, and consequently, Lagrange’s equation of i=1 · ∂q j motionP are unchanged under constrained motion. 4 of motion lead to 2s + 1 first order differential equation as follows. ∂H ∂H ∂L ∂H ✞ ☎ q˙i = ;p ˙i = ; = .

Load more