Physics-I Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: profanurag@gmail.com Visit me: Room-110, Block-E, IIITM Campus
Syllabus Electrodynamics: Maxwell s equations: differential and integral forms, significance of Maxwell s equations, displacement current and correction in Ampere s law, electromagnetic wave propagation, transverse nature of EM waves, wave propagation in bounded system, applications. Quantum Physics: Dual nature of matter, de-broglie Hypothesis, Heisenberg uncertainty principle and its applications, postulates of quantum mechanics, wave function & its physical significance, probability density, Schrodinger s wave equation, eigen values & eigen functions, Applications of Schroedinger equation.
Erwin Rudolf Josef Alexander Schrödinger Born: 1 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria Nobel Prize in Physics 1933 "for the discovery of new productive forms of atomic theory"
An equation for matter waves Seem to need an equation that involves the first derivative in time, but the second derivative in space Try t x ( xt, ) is "wave function" associated with matter wave As before try solution x, t e i kxt So equation for matter waves in free space is (free particle Schrödinger equation) i t m x
Schrödinger s Equation Solve this equation to obtain y Tells us how y evolves or behaves in a given potential Analogue of Newton s equation in classical mechanics Erwin Schrödinger (1887-1961)
Schrödinger s Equation This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree with experiment. There are limits to its validity. In this form it applies only to a single, non-relativistic particle (i.e. one with non-zero rest mass and speed much less than c) The Schrödinger equation is a partial differential equation in x and t (like classical wave equation). Unlike the classical wave equation it is first order in time. The Schrödinger equation contains the complex number i. Therefore its solutions are essentially complex (unlike classical waves, where the use of complex numbers is just a mathematical convenience).
Interpretation of y The quantity y is interpreted as the probability that the particle can be found at a particular point x and a particular time t The act of measurement collapses the wave function and turns it into a particle Neils Bohr (1885-196)
An equation for matter waves What about particles that are not free? Substitute gives x, t y For particle in a potential V(x,t) e i kxt into free particle equation i k m x, t y x, t Has form (Total Energy)*(wavefunction) = (KE)*(wavefunction) p E m V ( x, t) t m x p E m Total energy = KE + PE Suggests modification to Schrödinger equation: (Total Energy)*(wavefunction) = (KE+PE)*(wavefunction) i V ( x, t) t m x Schrödinger Time-dependent Schrödinger equation
The Schrödinger equation: notes This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree with experiment. There are limits to its validity. In this form it applies only to a single, nonrelativistic particle (i.e. one with non-zero rest mass and speed much less than c) The Schrödinger equation is a partial differential equation in x and t (like classical wave equation). Unlike the classical wave equation it is first order in time. The Schrödinger equation contains the complex number i. Therefore its solutions are essentially complex (unlike classical waves, where the use of complex numbers is just a mathematical convenience). Note the +ve sign of i in the Schrödinger equation. This came from our looking for plane waves of the form i t e We could equally well have looked for solutions of the form i t e Then we would have got a ve sign. This is a matter of convention (now very well established). i V ( x, t) t m x
The Hamiltonian operator Time-dependent Schrödinger equation i V ( x, t) t m x Can think of the RHS of the Schrödinger equation as a differential operator that represents the energy of the particle. This operator is called the Hamiltonian of the particle, and usually given the symbol Ĥ Hence there is an alternative (shorthand) form for the timedependent Schrödinger equation: i t Hˆ d V ( x, t) Hˆ m dx Kinetic energy operator Potential energy operator Hamiltonian is a linear differential operator. Schrödinger equation is a linear homogeneous partial differential equation
Interpretation of the wave function Ψ is a complex quantity, so how can it correspond to real physical measurements on a system? Remember photons: number of photons per unit volume is proportional to the electromagnetic energy per unit volume, hence to square of electromagnetic field strength. Postulate (Born interpretation): probability of finding particle in a small length δx at position x and time t is equal to * ( x, t) x Note: Ψ(x,t) is the probability per unit length. It is real as required for a probability distribution. Total probability of finding particle between positions a and b is Ψ δx Born b xa b ( x, t) x ( x, t) dx x0 a a b x
An equation for matter waves: the time-dependent Schrödinger equation Classical 1D wave equation v wave velocity 1 x v t e.g. waves on a string: ( xt, ) wave displacement ( xt, ) x Can we use this to describe matter waves in free space? Try solution x, t e i kxt But this isn t correct! For free particles we know that E p m
DOUBLE-SLIT EXPERIMENT REVISITED 1 y Incoming coherent beam of particles (or light) θ d sin D Detecting screen Schrödinger equation is linear: solution with both slits open is 1 Observation is nonlinear * * 1 1 1 Usual particle part Interference term gives fringes
Normalization Total probability for particle to be somewhere should always be one ( x, t) dx 1 Normalization condition A wavefunction which obeys this condition is said to be normalized Suppose we have a solution to the Schrödinger equation that is not normalized. Then we can Calculate the normalization integral Re-scale the wave function as (This works because any solution to the SE multiplied by a constant remains a solution, because the SE is LINEAR and HOMOGENEOUS) N ( x, t) dx 1 y( x, t) y( x, t) N New wavefunction is normalized to 1
Normalizing a wavefunction - example Particle with un-normalized wavefunction at some instant of time t (, ), x t a x a x a ( x, t) 0, x a
Conservation of probability If the Born interpretation of the wavefunction is correct then the normalization integral must be independent of time (and can always be chosen to be 1 by normalizing the wavefunction) ( x, t) dx constant Total probability for particle to be somewhere should ALWAYS be one We can prove that this is true for physically relevant wavefunctions using the Schrödinger equation. This is a very important check on the consistency of the Born interpretation. i V ( x, t) t m x ( x, t) dx constant
Boundary conditions for the wavefunction The wavefunction must: Examples of unsuitable wavefunctions 1. Be a continuous and single-valued function of both x and t (in order that the probability density is uniquely defined) y ( x) x Not single valued y ( x). Have a continuous first derivative (except at points where the potential is infinite) y ( x) Discontinuous x 3. Have a finite normalization integral (so we can define a normalized probability) x Gradient discontinuous
Time-independent Schrödinger equation Suppose potential is independent of time V x, t V ( x) LHS involves only variation of Ψ with t i V( x) t m x Look for a separated solution ( x, t) y ( x) T( t) RHS involves only variation of Ψ with x (i.e. Hamiltonian operator does not depend on t) Substitute: ( ) ( ) ( ) ( ) ( ) ( ) ( ) m x t y x T t V x y x T t i y x T t x ( x) T( t) T( t) d y y dx d dt T y V ( x) yt i y m dx dt etc N.B. Total not partial derivatives now
Divide by ψt 1 d y 1 V ( x) i dt m y dx T dt d dt T y V ( x) yt i y m dx dt LHS depends only on x, RHS depends only on t. True for all x and t so both sides must be a constant, A (A = separation constant) This gives i 1 dt T dt A 1 d y V ( x) m y dx A So we have two equations, one for the time dependence of the wavefunction and one for the space dependence. We also have to determine the separation constant.
SOLVING THE TIME EQUATION 1 dt i A T dt 1 dt i A T dt 1 d y V ( x) m y dx A dt dt ia T This is like a wave i t e T() t ae iat with A/.. So A E / T() t ae iet / This only tells us that T(t) depends on the energy E. It doesn t tell us what the energy actually is. For that we have to solve the space part. T(t) does not depend explicitly on the potential V(x). But there is an implicit dependence because the potential affects the possible values for the energy E.
TIME INDEPENDENT SCHROEDINGER EQUATION Consider a particle of mass m, moving with a velocity v along + ve X-axis. Then the according to de Broglie Hypothesis, the wave length of the wave associated with the particle is given by Erwin Schroedinger h mv A wave traveling along x-axis can be represented by the equation i t k x x, t A e 1
Where Ψ(x,t) is called wave function. The differential equation of matter wave in one dimension is derived as 0 8 y y V E h m dx d The above equation is called one-dimensional Schroedinger s wave equation in one dimension.in three dimensions the Schroedinger wave equation becomes 0 8 0 8 y y y y y y V E h m V E h m z y x
Time-independent Schrödinger equation With A = E, the space equation becomes: d y V ( x) y m dx Ey or Ĥy Ey Solution to full TDSE is This is the time-independent Schrödinger equation ( x, t) y( x) T( t) y( x) e iet / Even though the potential is independent of time the wavefunction still oscillates in time But probability distribution is static * iet / iet /, y, y ( ) y ( ) P x t x t x e x e * y y y ( x) ( x) ( x) For this reason a solution of the TISE is known as a stationary state Solving the space equation = rest of course!
Notes d y V ( x) y Ey m dx In one space dimension, the time-independent Schrödinger equation is an ordinary differential equation (not a partial differential equation) The time-independent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator: Operator function = number function (Compare Matrix vector = number vector) Ĥy Ey We will consistently use uppercase Ψ(x,t) for the full wavefunction (TDSE), and lowercase ψ(x) for the spatial part of the wavefunction when time and space have been separated (TISE)
Energy of the electron Energy is related to the Principle Quantum number, n. This gives 3 of the 4 quantum numbers, the last one is the spin quantum number, s, either +½ or ½.
Wave Functions: Probability to find an electron
Energy of the electron
Electron Transitions give off Energy as Light/Xrays E=hc/
Red/Blue Clouds in Space
Zeeman Effect Light Emission in Magnetic Field
Multi-electron Atoms (Wolfgang) Pauli Principle Exclusion Principle No electrons with same quantum numbers!
Energy Eigenvalues
Eigen value and Eigen Function
Applications:
Operators in Quantum Mechanics