Presentation on theme: "“velocity” is group velocity, not phase velocity"— Presentation transcript:
1“velocity” is group velocity, not phase velocity Quantum WavesParticles are actually wavesTwo de Broglie relationsThey have all the properties (and equations) that described waves before, together with some new onesRewrite the de Broglie relations in terms of k and :We can also find a dispersion relation for these waves:From this, we can find the group and phase velocity:“velocity” is group velocity, not phase velocity
2Quantum Uncertainty Two uncertainty relations for any type of wave Multiply by h-barUse quantum wave relationsUncertainty in Position/MomentumClassical physics:Particle is defined by position x and velocity v = (dx/dt)Or, if you prefer, exact x and exact pQuantum physics:Particle is defined by wave function (x)It cannot have a definite position and momentum – they tend to both be uncertainUncertainty in Time/EnergyThings that last eternally can have definite energyThings that last a brief time have uncertain energyMeasurements of energy will have a spread in value
3Time/Energy Uncertainty The 0-meson has a rest mass of MeV/c2 and lasts an average of 8.410-17 s. What is the spread in energies of a 0-meson at rest due to its finite lifespan?EEE - mc2 (eV)
4These methods give only estimates of the answer Consequences of Position UncertaintyPosition and momentum of a particle cannot be simultaneously specified or determinedWe can often estimate one quantity if we know the other by treating this inequality as if it were an equalityEspecially when the energy is being minimizedLEnergy of Particle in a BoxBy Carlson’s rule, the position is uncertain by about x = ¼L.By the uncertainty principle, momentum is uncertain byWe’d like the momentum to be zero, but we can’tThis causes the particle to have some energy, called Zero Point Energy?These methods give only estimates of the answer
5Solving Uncertainty/Energy Problems A particle of mass m lies above an impenetrable barrier in a gravitational field with acceleration g. What is the minimum energy of the particle?yWrite an expression for the energy: potential and kineticFigure out what momentum (normally 0) and what position would have the “ideal” lowest kinetic and potential energiesLet x = a and p = /2aAssume the momentum and position differ from ideal by about x = a and p =/2aSet derivative of energy function equal to zero, solve for aSubstitute a in to determine minimum energy
6Uncertainty and the Hydrogen Atom There are two types of energy associated with the Hydrogen atomPotential EnergyKinetic EnergyClassically, these two energies are at a minimum when:The electron is at rest, p = 0The electron is at the origin, r = 0 (x = 0)In this case, the energy of the hydrogen atom E = –Quantum mechanically, you can’t control both position and momentumWe will place the electron near the origin, r = 0, but there is an uncertainty x associated with itWe will place the electron nearly at rest, p = 0, but there is an uncertainty p associated with itYou have to compromise between these two choices to minimize ELet x = a, and let p be as small as possible by the uncertainty principlePlug these into the formula for energy
7Uncertainty and the Hydrogen Atom We need to minimize the energyTake derivative and set it to zeroSubstitute result back inCorrect answer is 4 times smaller than this. Why so far off?It’s just an estimateThere is momentum in three dimensionsThis makes answer 3 times smallerThen answer we get is 4/3 of correct answer
8“Electric Field is potential for force at a distance on a charge”. The Wave FunctionWe’ve been talking about waves, which require wave functionsWe’d better give it a name:It would have more arguments in more dimensionsIt is a complex function; it has both a real and imaginary partsWhat does it mean?This turns out to be very hardAnalogy: Electromagnetic WavesDescribed by electric and magnetic fieldsWhat is an electric field?It is defined in terms of a potential for somethingAmerican Heritage Science Dictionary“The distribution in space of the strength and direction of forces that would be exerted on an electric charge at any point in that space”.Dr. Carlson, PHY 114“Electric Field is potential for force at a distance on a charge”.
9The Wave Function: what does it mean? We are talking about one particle – but it is not at one location in spaceIf we measured its position, where would we be likely to find it?The Wave Function is also called the probability amplitudeClearly, where the wave function is small (or zero), you wouldn’t expect to find the particleWhere it’s negative or imaginary, wouldn’t expect to have negative or imaginary probabilityWe’d better make darn sure that the probability is always positiveFor electric fields, the energy density is proportional to the field squaredIf working with complex waves, take amplitude firstHow about we make probability density proportional to wave function magnitude squared:
10Sample ProblemA wave in the region 0 < x < a has the wave function above. What is the probability density at all locations x at all times t?
11Sample ProblemA wave in the region 0 < x < a has the wave function above. What is the probability density at all locations x at all times t?
12… in the region 0 < x < a … What Does Probability Density Mean?The probability density (in 1D) has units of m-1In a small region of size dx, the probability of finding the particle is there is given by ||2dx.To find probability over a larger region, you have to integrate itNormalization: The probability that the particle is somewhere must be 1If we integrate over all x, we must get 1In some cases, the problem implies that we restrict to some region… in the region 0 < x < a …
13Sample ProblemAt t = 0, the wave function is given by the expression below.What is the normalization constant N?What is the probability that the particle is at x > ½a?
14Sample ProblemAt t = 0, the wave function is given by the expression below.What is the normalization constant N?What is the probability that the particle is at x > ½a?
15Sample ProblemAt t = 0, the wave function is given by the expression above.What is the most likely / least likely places to find the particle?What is the normalization constant N?What is the probability that the particle is at 0 < x < a?Least likely when function vanishes, at x = 0Most likely when function is largest positive or negativeNormalization:Let x = atan
16Sample ProblemAt t = 0, the wave function is given by the expression above.What is the most likely / least likely places to find the particle?What is the normalization constant N?What is the probability that the particle is at 0 < x < a?
17End of material for Test 2 Quantum Wave Equations You Need:End of material for Test 2