1. 1AMQ P.H. Regan 1 1AMQ, Part II Quantum Mechanics 4 Lectures n Wave Particle Duality, the Heisenberg Uncertainty Principle, de Broglie (wavelength) hypothesis n Schrodinger Wave Equation. n Wavefunctions and the Free Particle. Electrons in a box, energy quantization. K.Krane, Modern Physics, Chapters 4 and 5 Eisberg and Resnick, Quantum Physics, Chapters 3 (and 5 & 6)

2. 1AMQ P.H. Regan 2 Wave-Particle Duality. Light behaves like: (a) a wave (diffraction, interference) and/or (b) a stream of massless particles or photons, (black body spectrum, photoelectric effect, Compton scattering). Consider diffraction from a slit, one photon at a time. The experiment shows that individual particles of light gradually build up the diffraction pattern predicted by the classical wave theory. The wave pattern describes the probability of detecting a photon at that point. The wave pattern measures the probability detecting a photon at that point.

3. 1AMQ P.H. Regan 3 Individual photons unpredictable in detail, BUT their average behaviour is predictable! At any point P in the interference pattern, the intensity gives the probability: ie I(P) is prop. to prob. that a photon hits pos. P. For waves, the intensity = (amplitude) 2 and thus the wave and particle pictures are connected, Probability  amplitude of wave) 2 This is called the Born Approximation (Max Born, Nobel Prize, 1954). The question then arises, if waves can act like particles, is the reverse true ? The de Broglie Hypothesis (Nobel Prize, 1929) Louis de Broglie (for his PhD thesis!) postulated that in some phenomena, particles can act like waves!

4. 1AMQ P.H. Regan 4 He guess that for particles, the momentum, p, could be related to a wavelength,, by p = h/  where  is the de Broglie wavelength. Note this is consistent with the momentum for photons, ie p = E/c = h  h/ This predicts, for example, that electrons should show diffraction effects (like photons), if they pass through a slit with width ~ dB. Calculating the de Broglie Wavelength. Example. e - s in a TV K e- =20 keV, m e =9.1x10 -31 kg, gives dB =0.009nm. (Note that the typical atomic spacing is ~0.05nm). dB =0.05nm corresponds to K e- =601 eV for electrons. [from K e =p 2 /2m e =(1/2m). (h/ ) 2 ] ie. e - s accelerates across 600 eV should be diffracted exactly like x-rays with  0.05nm.

5. 1AMQ P.H. Regan 5 X-ray diffraction pattern Electron diffraction pattern

6. 1AMQ P.H. Regan 6 Electron and neutron diffraction are widely used in crystallography studies. Note how (2 slit) e - interference pattern builds up with increasing number of e - s.

7. 1AMQ P.H. Regan 7 The y-position for any transmitted photon is specified to an accuracy (uncertainty)  y=a. Making the slit smaller (ie. reducing  y) increases the uncertainty in the angle, 

8. 1AMQ P.H. Regan 8 Thus,  y  p y ~ h (for single slit diffraction). This relates to the physical limits we can put on determining either the position or momentum of the individual photons as they pass through the slit. These are a subset of the Heisenberg Uncertainty Principle. In general, if p = momentum, t = time and E= energy, h = Planck’s constant,then, The division by 2  comes about due to the use of wave number (k) rather than wavelength (  k) in the definition of momentum (see Krane p114).

9. 1AMQ P.H. Regan 9 ie. given V(x), solve for x(t) or V(x,t). The Schrodinger Equation. For particles, forces change momentum (p=mv) and each force has an associated potential energy.  ewton’s 2nd law related potential to the change in momentum, via, For matter waves, p=h/  Thus, we need a differential equation to calculate how forces (described by a potential energy) affect or p. We will solve this problem for a wave, which has an amplitude (  for each value of position (x) and time (t).

10. 1AMQ P.H. Regan 10 Mathematical Forms of Waves.  x,t)=    cos(kx-  t), k=2  and  We can re-write this formula in terms of e, as  x,t  o e i(kx  t    e  i  kx  t   Recalling e ix = cosx + isinx, the simplest form of a wave is  x,t  o e i(kx  t  Formulation of the Schrodinger Equation  o relate p and V we use, Total Energy (E) = Kinetic (K) + Potential (V)

11. 1AMQ P.H. Regan 11 Now, if Thus, This is the Schrodinger equation.

12. 1AMQ P.H. Regan 12 Using the Schrodinger Equation. (i) specify V=V(x,t) (ie details of force) (ii) specify initial conditions (ie. at t=0) and (iii) specify boundary conditions (x=fixed values)

13. 1AMQ P.H. Regan 13 In many cases, (eg. Coulomb potential in H atom), V=V(x), ie. not time dependent, then where  x) satisifies Time Independent Schrodinger Equation Properties of the T.I.S.E. Applies when V=V(x), ie. V is independent of time.Eg. an e - wave acted on by a fixed nucleus, but NOT an atom in an oscillating magnetic field. Applies only in the non-relativistic limit (ie. assumes K=(1/2)mv 2 =p 2 /2m). Assumes p=h/ and E=h. Is linear in  ie no terms like     etc. Laws of superposition are valid. If  a and  b are two possible solutions, then  c a  a +c b  b is a solution for all constants c a and c b.

14. 1AMQ P.H. Regan 14 Wavefunctions We have 2 forms of Schrodinger equation, (a) general, time-dependent solve for  (x,t), when V(x,t) is given, and (b) TISE, for V=V(x) only. Solve for  x) when V(x) is given. Solution is a standing wave, ie.  (x,t)=  (x) e -i  t Wavefunctions are solutions to the Schrodinger wave equation. The w.function,  x,t) describes physical state of the particle, such as its momentum, energy etc. and also where the particle (probability wise) is.

15. 1AMQ P.H. Regan 15 The Born Approximation As in the earlier electron and photon diffraction examples, the wave intensity is intepreted as giving the probability of finding the particle in a particular state. Recalling that the Intensity =|Amplitude| 2, we interpret |  (x,t)| 2 dx =probability the particle will be found between x and x+dx at time t. Solving the Schrodinger equation specifies  (x,t) completely, except for a constant, ie. if  is a solution, so is A x  From the Born approx. the normalisation condition of the wavefunction is that ie. The probability that particle is somewhere = 1 Note that probability is a real number. Although  is complex,  |  2 | is real.

16. 1AMQ P.H. Regan 16 Solution of the Wave Equation for a Free Particle `Free particle’, means V=0 everywhere, such as an electron in the absence of external forces. Using the TISE, The energy is given by H , thus Since k (the wavenumber) is real, we have shown that any positive value of energy is allowed for a free particle. Try a solution of the form,  e ax,

17. 1AMQ P.H. Regan 17 There are also boundary conditions that  (x) = 0 for x=0 and x=L, for all times, t Particles (electrons) in a Box: Solving the Wavefunction for Bound Electrons. When electrons are acted on by a force, this acts to restrict the allowed energies (and hence other quantities eg. momentum, spin..) This force is specified by the potential, V(x,t). Example: Electrons in a 1-D Box Suppose an e - (in 1-d) is placed in a `box’ with impenetrable walls at x=0 and x=L. The electron can move freely (in x direction) in the box, such that: V=0 x=0 x=L Note if V tends to infinity, the prob. of finding the e - there must tend to zero.ie  >0 outside box.

18. 1AMQ P.H. Regan 18 Outside the box,  =0, now we solve for inside. Since inside the box, V=0 for all t, we can use the T.I.S.E, ie. For a free particle, we saw that Ae i( k x) was a solution for any value of k and that For a particle in a box, we consider the most general solution for a given energy, and then apply the boundary conditions. General solution is of the form Thus we can write,

19. 1AMQ P.H. Regan 19 One of the boundary conditions is  (0)=0, thus, B=0 and hence,  (x)=Asin(kx). We also have  (L)=0, so 0=Asin(kl), hence, either, A=0 or sin(kL)=0. If A=0, the wavefunction =0 everywhere! (ie. there is no probability of the a particle having this wavefunction!(solution can be disregarded). Hence, kL = n , where n=1,2,3…., ie k=n  L and thus Determine A using the normalisation condition, Thus BCs mean that only certain modes are allowed for  (x,t). A plot of the real (or imag.) part of  (x,t) looks like standing waves.

20. 1AMQ P.H. Regan 20

21. 1AMQ P.H. Regan 21

22. 1AMQ P.H. Regan 22 Energy Quantization For particles (eg, electrons) in box, the boundary conditions mean that only certain modes are allowed for the wavefunction, ie. which arises from from the restriction, But the particle energy is given by, The restriction on k, means only certain energies are allowed, ie. the energy is quantized. Thus, by simple substitution, the energy levels are given by,

23. 1AMQ P.H. Regan 23 Note the ZPO is consistent with the uncertainty principle, ie. particles with zero momentum can not be localised.