More on waves and Uncertainty Principle

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Reynold Gardner
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1 More on waves and Uncertainty Principle Announcements: No Class on Friday Homework not due until Thursday. Still have several homework papers to pick up (get at end of hour) Physics 2170 Fall

2 Homework # 5 Physics 2170 Fall

3 Sum of Homeworks Physics 2170 Fall

4 Clarification of de Broglie relations The de Broglie relation is Originally came from an analysis of massless photons but also works for massive particles like electrons, neutrons, and atoms. In fact, there is another relation which is derived from the photon results: Note the momentum relation deals with the space part of a wave (wavelength and wave number) while the energy relation deals with the time part of the wave (frequency). For light, the space and time quantities are related by For massive particles but so it is not very useful in practice. My advice: avoid using velocity. Stick with E, p, k, T, f, λ, ω. Physics 2170 Fall

5 Plane wave: Wave packet: Plane Waves vs. Wave Packets This wave represents a single k and ω. Therefore energy, momentum, and wavelength are well defined. The amplitude is the same over all space and time so position and time are undefined. This wave is composed of many different k and ω waves. Thus, it is composed of many different energies, momenta, and wavelengths and so these quantities are not well defined. The amplitude is non-zero in a small region of space and time so the position and time is constrained to be in that region. Physics 2170 Fall

6 Heisenberg Uncertainty Principle: Δx small Δp only one wavelength Δx medium Δp wave packet made of several waves Δx large Δp wave packet made of lots of waves Physics 2170 Fall

7 Heisenberg Uncertainty Principle: Δt small ΔE only one period Δt medium ΔE wave packet made of several waves Δt large ΔE wave packet made of lots of waves Physics 2170 Fall

8 Heisenberg Uncertainty Principle There are two Heisenberg uncertainty relations: Really 3 in space Δx Δp x ħ/2 ; Δy Δp y ħ/2 ; Δz Δp z ħ/2 If angular coordinates, then Δθ ΔL θ ħ/2 The wave nature of things prevents a precise determination of both momentum and position or of both energy and time. This is a fundamental limitation which has nothing to do with the actual equipment used to measure things. Another way of seeing why this makes sense is Heisenberg s microscope. will discuss in a few slides Microscopes are limited to resolutions ~ wavelength of light Smaller wavelengths allow a better measurement of x but the photons have larger momentum giving larger kicks to the particle, making the momentum more uncertain. Physics 2170 Fall

9 Heisenberg Uncertainty Comments In classical physics it is taken for granted that particles have definite values for their position, x, and momentum, p. It was recognized that x and p could not be measured with perfect accuracy, but sort of assumed that with care one could make the uncertainties as small as one wanted! Heisenberg s uncertainty relation shows the assumptions were incorrect. Either uncertainty on momentum or position can be made as small as one wants, but the product can never be less than ħ/2. The uncertainty principle applies to all particles, but on the macroscopic scale the limit is so small that it is just not recognizable. Physics 2170 Fall

10 Particle confined to a region Take the example of an electron confined to an atom with total width of a 0.1nm. (like the Hydrogen atom) As measured from the center Δx a/2 From the uncertainty principle Δp ħ/2δx ħ/a Δp = m Δv or Δv = Δp/m ħ/am Multiplying top and bottom by c 2, gives: Δv (ħc) c/a mc 2 = 197 ev-nm c/(0.1nm.511mev) c/ m/s Large uncertainty in v shows great importance of uncertainty principle for systems having atomic dimensions Physics 2170 Fall

11 Zero-point Energy Particle confined to a space cannot be at rest! Δp ħ/a ; momentum is spread out by at least this amount <Kinetic Energy> = <K>= <p 2 /2m> (Δp) 2 /2m <K> ħ 2 /2ma 2 = (ħc) 2 /(2 mc 2 )a 2 = (197eV nm) 2 /[(1 MeV)(0.1nm) 2 ] = 4 ev Note this bound is lower than the 13.6eV kinetic energy of the electron in the hydrogen ground state Zero point energy is the minimum possible kinetic energy for a particle confined to a region of space. Physics 2170 Fall

12 Atomic Clocks Uncertainty Principle ΔΕ is the uncertainty in a particle s energy a quantum particle does not have a definite energy! Δt describes the time where the particle is likely to be found at position x. If a particle has a definite energy, then ΔΕ=0 and the particle stays in the same state (position) forever these are called stationary states! If a particle does not remain in the same state forever, then Δt is finite and miinimum uncertainty is ΔΕ ħ/2δt Physics 2170 Fall

13 Atomic Clocks cont. ΔΕ ħ/2δt Typical time for a excited to decay by Emission of a photon is 10-8 s. ΔΕ ħc/2 c Δt = 197 ev nm/[2 (3 x 10 8 m/s) (10-8 s)] ev The frequency of photons ejected in atomic transitions are used as standards for definition (and calibration) of frequency and time. Because of uncertainty principle the frequency of photon is uncertain by Δω = ΔΕ/ħ 1/(2 Δt) -- states with very long Δt are chosen as standards. Physics 2170 Fall

14 Clicker question 1 Set frequency to AD A photon can turn into an electron-positron if there is sufficient energy. If there is insufficient energy, then the electron-positron pair can live on borrowed time for about how long? A. 6 x s B. 12 x s C. 3 x s D. 2000s E. None of the above ħ= ev s m e c MeV Physics 2170 Fall

15 Clicker question 1 Set frequency to AD A photon can turn into an electron-positron if there is sufficient energy. If there is insufficient energy, then the electron-positron pair can live on borrowed time for about how long? A. 6 x s B. 12 x s C. 3 x s D. 2000s E. None of the above ħ= ev s m e c MeV Δt = ħ/2δε = 6.58 x ev s/(2 x 10 6 ev) 3.3 x s Physics 2170 Fall

16 Clicker question 2 Set frequency to AD The positions of two objects are measured with the same accuracy while their velocities are measured as accurately as the uncertainty principle allows. If one object has ten times the mass of the other, then the more massive object has a velocity uncertainty which is A. 1/100 B. 1/10 C. the same as D. 10 times the velocity uncertainty of the less massive object Physics 2170 Fall

17 Clicker question 2 Set frequency to AD The positions of two objects are measured with the same accuracy while their velocities are measured as accurately as the uncertainty principle allows. If one object has ten times the mass of the other, then the more massive object has a velocity uncertainty which is A. 1/100 B. 1/10 C. the same as D. 10 times the velocity uncertainty of the less massive object Δv = ħ/2m Δx uncertainty in velocity scales as 1/m Physics 2170 Fall

18 Uncertainty Principle Application Consider a particle of mass m which is confined to a three dimensional box of side length L but free to move within the box. Use the uncertainty principle to estimate the minimum value of average kinetic energy for such a particle. How do we solve this problem? - first find the uncertainty in x previously we used L/2, but we can do better using rms Δx = x 2 x 2 = x 2 x 2 = 1 L L / 2 x 2 dx = 1 L (x 3 /3) L / 2 L / 2 = (L 2 /24) ( L 2 /24) = L 2 /12 L / 2 Δx = L 2 /12 = L / 12 Physics 2170 Fall

19 Uncertainty Principle Application cont. The uncertainty in momentum is The average kinetic energy can be expressed in terms of the average of the momentum squared, which is related to the uncertainty in momentum by The average momentum is zero for free random motion of the particle in a box. The y and z coordinates are equivalent, so the average kinetic energy can be expressed as Physics 2170 Fall

20 Estimate the H Ground State Energy We can use the uncertainty principle to estimate the minimum energy for Hydrogen. The idea is that the radius must be larger than the spread in position, and the momentum must be larger than the spread in momentum. Let s take rp ħ Total Energy = E = p 2 /2m - ke 2 /r = p 2 /2m - ke 2 p/ħ Differentiate E wrt p and set equal to zero to get the minimum. de/dp = p/m ke 2 /ħ = 0 p =mke 2 /ħ E = -1/2 m(ke 2 ) 2 /ħ 2 = ev Physics 2170 Fall

Chapter 38 Quantum Mechanics

Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 38-3 The Heisenberg Uncertainty Principle