Matter Waves. Byungsung O Department of Physics, CNU x6544

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1 Matter Waves x6544

2 Electrons and Matter Waves If electromagnetic waves (light) can behave like particles (photons), can particles behave like waves? forwaves, then forparticles (de Broglie wavelength) x-ray diffraction electron diffraction

3 De Broglie wavelength What is the de Broglie wavelength for anelectron with =120 ev? = MeV : non-relativistic ㆍ kgㆍm/s m =0.112 nm = 112 pm

4 Wavefunctions To describeamatterwave, wavefunction In all the situation, where ()is theangularmomentum What does the wavefunction mean? A matterwaveisaprobabilitywave: theprobability that a particle will be detected in a specific time interval is proportional to (which isreal and non-negative) ( ):probabilitydensity

5 How do we find the wavefunction? Sound waves / waves on strings: Newton's equation Light waves: Maxwell's equation Matter waves: Schro"dinger's equation Schro"dinger's equation ( )

6 Schro"dinger's equation for a free particle Freeparticle: and or The general solution with travelling travelling +x direction -x direction [Any ± representsatraveling wave]

7 Wavefunction for a free particle Assumeafreeparticleis movingin thepositivedirection of The probability density (aconstant) Prob. density is thesameforall. Theparticlehas equal prob. of being anywherealong axis

8 Uncertainties In the previous example, the momentum is exactly defined but the position is completely unknown. In general, and Heisenberg's Uncertainty Principles W. Heisenberg in 1927 If,then,, and.

9 Uncertainty Principle The velocity of an electronmoving along x-axis = 2.05x10 6 m/s.and the uncertainty is 0.50 %. What is the minimum uncertainty when the position is measured simultaneously? =(9.11x10-31 )(2.05x10 6 ) = 1.87x10-24 kgㆍm/s = (0.0050)(1.87x10-24 ) = 9.35x10-27 kgㆍm/s = 1.13x10-8 m = 11.3 nm

10 Potential step 1 incoming wave 1. At, 2.1For,at with Thewavemusttravel in direction

11 Properties of the wavefunctions 1. The wavefunction should be continuous 2. The first derivative of the wavefunction should be continuous Namely,at theinterface( ) and

12 Potentialstep 2(for ) Soat Reflection: and : even, non-zeroprob. ofturning back! (optical analogy) :theparticlewill bereflectedortransmitted Transmission: goes1as

13 Potential step 3 2.2For,at with Thewavefunction shoulddecay inthebarrier At and

14 Potentialstep 4 (for ) Reflection As in C.M., the particle is always reflected(total reflection). But non-zero prob.( ) ofpresenceintheclassically forbidden region!! iscomplex:phaseshift upon reflection

15 Potential barrier 1 For 1)for : 2)for : 3)for : with and At, at,

16 Potentialbarrier2(for ) and again! :oscillates!! when (i.e. ), : resonance

17 Transmission Probability ev, nm ev, nm

18 Potentialbarrier3(for ) for for (for ) for with and

19 Potentialbarrier4(for ) Byreplacing in theaboveresultswith when, and where Non-zero prob. of crossing the potential barrier: tunneling

20 Tunneling Probability In general, the transmission probability is given by where and arethe2classical turning points for the potential barrier.

Asaresult, thetip maps outthecontoursofthesurfacewith resolutiononthescaleof1nminstead of >300 nm for optical

21 Tunneling alpha-decay scanning tunneling microscope(stm) As tip is scanned laterally across the surface, thetipis moved up or down to keep the tunneling current(tip to surface distance ) constant. Asaresult, thetip maps outthecontoursofthesurfacewith resolutiononthescaleof1nminstead of >300 nm for optical microscopes! 92U Th He ++ () From the tunneling prob., the lifetime can be estimated.

22 STM images Blue Platinum Quantum Corral Pt(111) Iron on Copper(111)

23 Tunneling Probability Anelectron with = 5.1 ev approaches to asquare barrier with the height of 6.8 ev and the thickness of 750 pm. What is the probability to find the electron tunneling through? where = 6.67x10 9 m -1 = 10.0 =4.5x10-5. The energy of the electronpassing through=5.1 ev What is the probability for a protonwith =5.1 ev? : mathematically not the same as 0 but practically almost same as 0

Final Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall.

Final Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the