Quantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :

Size: px

Start display at page:

Download "Quantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :"

Osborn Glenn
5 years ago
Views:

1 Student Selected Module 2005/2006 (SSM-0032) 17 th November 2005 Quantum Mechanics Outline : Review of Previous Lecture. Single Particle Wavefunctions. Time-Independent Schrödinger equation. Particle in a Box. Quantum Superposition The Hydrogen Atom. 1 Review of Previous Lecture Wave-particle duality : all objects display both wave and particle properties The De Broglie equation relates wave and particle properties : p = h /" The Uncertainty Principle places fundamental limits on our measurements : p " = momentum = wavelength "x "p x # h 2 "E "t # h 2 Quantum mechanical waves are "probability waves" : Probability " # 2 " is generally complex. We want to look at the equations governing these quantum mechanical probability waves. 2

2 Single Particle Wavefunctions In 1-dimension : "(x) "x x Probability for the particle to be found in the small interval!x is : "(x) 2 # $x Given that the probability to find the particle anywhere must be exactly 1, we have the condition : # "(x) 2 dx =1 Wavefunction normalisation 3 Schrödinger Equation (in 6 easy steps) 1) Consider the following simple wavefunction for a particle : "(x) = Acos(kx) [ k = 2" /#] $ A x 2) Since $(x) extends across all space (x), it provides no information about the position of the particle. 3) Therefore, by the uncertainty principle, it described a particle of precisely known momentum. 4) For particles of known momentum : E = K.E.+ P.E. = p2 2m + V = h2 k 2 2m + V Wavelength " = 2#/" De Broglie : p = h /" = hk 6) Multiply by coskx and derivatives : E coskx = "h2 2m d 2 coskx dx 2 + V cos kx "h 2 2m d 2 #(x) dx 2 + V#(x) = E#(x) Time-independent Schrödinger equation. 4

3 Schrödinger Equation A lot of quantum mechanics consists in simply solving this equation for different potentials V : Potential : Physical significance : 1 2 Kx 2 q 1 q 2 4"# 0 r Complicated crystal lattice Simple harmonic oscillator Coulomb potential : atoms Solid state physics 5 Particle in a Box We're going to start with something much simpler - a particle in a one dimensional box. The box is represented by an infinite potential well. V = % V (potential energy) V = % V = 0 0 L x Wavefunction must be 0 in these regions, otherwise there would be a finite probability to find the particle with infinite energy. 6

4 Particle in a Box For the region 0 & x & L : ' "h 2 2m d 2 #(x) dx 2 = E#(x) "(x) = Acoskx + Bsinkx 1 Since "(0) = "(L) = 0 this leaves : $ "(x) = N sin n#x ' & ) n =1,2,3,... % L ( Normalisation factor Substituting in 1 gives : E = h2 " 2 n 2 2mL 2 A discrete set of allowed energy levels : quantisation of energy. 7 Particle in a Box "(x) "(x) 2 8

Particle in a Box "(x) 2 As n increases, the probability distribution approaches the classical expectation : uniformly distributed across the width of the box ' correspondence principle.

5 Particle in a Box "(x) 2 As n increases, the probability distribution approaches the classical expectation : uniformly distributed across the width of the box ' correspondence principle. 9 Particle in a Box E = h2 " 2 n 2 2mL 2 n = 4 n = 3 Energy Level Diagrams : Represent allowed energy levels ("energy eigenvalues"). Labelled by quantum numbers (in this case just n). Interactions can be represented as transitions between energy states. Similar diagrams for other observables, for example angular momentum. n = 2 n = 1 0 Lowest energy state (ground state) is not zero. This is a common feature of quantum mechanical systems that is not generally the case classically. 10

= #E 1 t /h Animation 11 Time Dependence Particle in a box : n = 2 h = m = L = 1 ' E 2 = 4"

6 Particle in a box : n = 1 h = m = L = 1 ' E 1 = " 2 2 Time Dependence complex phase = $ rotates with time "(x,t) = ( Et /h) $ 2 sin #x ' & ) e *ie 1t / h % L ( IMAGINARY 0 REAL " = #E 1 t /h Animation 11 Time Dependence Particle in a box : n = 2 h = m = L = 1 ' E 2 = 4" 2 2 $ "(x,t) = 2 sin 2#x ' & ) e *ie 2 t / h % L ( IMAGINARY 0 REAL " = #E 2 t /h Animation 12

7 Quantum Superposition If " 1 and " 2 are both solutions to the Schrödinger equation, then so is : " # A"1 + B"2 The wavefunction must still be correctly normalised to give unit probability : " # " 1 " 2 " 1 A 2 + B 2 (A" 1 + B" 2 ) and are solutions corresponding to energies E 1 and E 2. What energy does a particle described by have? It does not have a well defined energy. An energy measurement could yield E 1 with probability ( A 2 or energy E 2 with probability ( B 2. Note : to show that this is the correct way to normalise the combined wavefunction, you will need to know that when both wavefunctions are real : $ " 1 #" 2 dx = 0 13 This is fairly easy to show for the wavefunctions describing a particle in a box that we have already seen. Quantum Superposition The average energy obtained in an energy measurement is still well defined : E = 1 (A 2 + B 2 ) (A2 E 1 + B 2 E 2 ) One could also define the spread or standard deviation of a series of energy measurements on the same state ". This is how one formally derives the quantum mechanical uncertainty relations. 14

8 Quantum Superposition Particle in a box : h = m = L = 1 " = 1 2 (" 1 +" 2 ) E = h2 " 2 n 2 2mL 2 n = 3 n = 2 n = 1 0 E Animation 15 Quantum Superposition Particle in a box : h = m = L = 1 " = (0.8" " 2 ) E = h2 " 2 n 2 2mL 2 n = 3 n = 2 n = 1 0 E Animation 16

Quantum Superposition Particle in a box : h = m = L = 1 " = 1 0.68 (0.2" 1 + 0.

: h = m = L = 1 " = 1 6 (" 40 +" 41 +" 42 +" 43 +" 44 +" 45 ) "(x) 2 Motion begins to look classical.

9 Quantum Superposition Particle in a box : h = m = L = 1 " = (0.2" " 2 ) E = h2 " 2 n 2 2mL 2 n = 3 n = 2 E n = 1 0 Animation 17 Quantum Superposition Particle in a box : h = m = L = 1 " = 1 6 (" 40 +" 41 +" 42 +" 43 +" 44 +" 45 ) "(x) 2 Motion begins to look classical. Time averaged probability is uniform in x, as would be expected for a classical particle rattling around in a box. Animation 18

10 Schrödinger s Cat We know that quantum superpositions exist on microscopic scales since we observe the resulting interference effects. Devise a scheme to transfer this microscopic superposition into a macroscopic one : Acknowledgement : Ard Louis 19 Schrödinger s Cat 20

11 The Hydrogen Atom Classical electromagnetism does not predict a stable atom : Electron suffers centripetal acceleration and radiates energy in the form of electromagnetic waves. Eventually the electron would spiral into the nucleus. In general, quantum mechanical systems do not have a ground state corresponding to the classical energy minimum (e.g. particle in a box). In the case of the hydrogen atom, the quantum mechanical ground state corresponds to an electron at a radius of : a 0 " #10 $10 m (Bohr radius) Since there are no lower energy states to occupy, the electron cannot lose more energy through radiation ' the hydrogen atom is stable! 21 The Hydrogen Atom The quantum mechanical prescription for understanding the hydrogen atom is simple. We just find the solutions to the Schrödinger equation with the relevant potential : "h 2 2m $ & % d 2 # dx + d 2 # 2 dy + d 2 # ' 2 dz 2 ) " e2 ( 4*+ 0 r # = E# Coulomb potential r = x 2 + y 2 + z 2 " ="(x, y,z) The approach is exactly the same as for a simple 1-dimensional problem such as a particle in a box, but the algebra is a bit more complicated. Energy of electron defined by principal quantum number n = 1, 2, 3,... (for a free atom in the absence of external magnetic fields etc.) 22

12 Probability The Hydrogen Atom r /a 0 23 The Hydrogen Atom Free electron has zero energy Transitions result in a photon emitted with energy corresponding to the gap. Ground State 24

13 Hydrogen Atom Wavefunctions pure 6,4,1> state : equal superposition of 4,3,3> and 4,1,0> states : equal superposition of 3,2,2> and 3,1,-1> states: Acknowledgement : Dauger Research 25 Summary We have looked at some of the laws that dictate the behaviour of quantum mechanical "probability waves". Most non-relativistic quantum mechanical systems can be understood just by solving the Schrödinger equation for the relevant potential. It always emerges that bound states (e.g. a particle in a box or an electron in an atom) have a discrete energy spectrum ) energy is quantised. Loosely speaking, this is because we have to fit a certain number of De Broglie wavelengths into the space available (like standing waves on a violin string). Different quantum states can be superposed on top of one another. In that case the energy of the system might not even be well defined. Superposition : o exists for macroscopic systems? (Schrödinger s Cat) o is what enables us to exploit quantum mechanics : quantum computing, quantum teleportation, etc. 26

14 Exercises 1 The figure below shows the wavefunction for a particle in a finite potential well, with energy between V 1 and V 2 : i. Why is it acceptable for the wavefunction to be negative in certain regions? ii. Can the particle escape from the potential well classically? iii. Can the particle escape from the potential well quantum mechanically? iv. What physical situations might such a potential represent? potential wavefunction 27 Schrödinger Equation (more advanced derivation) A suitable representation of a travelling wave is : D(x,t) = Acos(kx "#t) k = 2" /# $ = 2" /% Displacement Amplitude Distance/ Time Wavelength " or Period * The quantum mechanical equivalent of the plane wave is simply : "(x,t) = Ae i(kx#$t ) = A[cos(kx #$t) + isin(kx #$t)] This has the property that : "(x) 2 = A 2 This is completely independent of x This wavefunction contains no information about the location of the particle. But if!x is infinite, then by the uncertainty relationship, the momentum is known perfectly. 28

15 Schrödinger Equation (more advanced derivation) So, the following wavefunction describes a particle of unique momentum : This means that I expect the following two relations to be relevant : De Broglie : Kinetic energy : p = hk These are both satisfied if the above wavefunction is inserted into : "h 2 2m "(x,t) = Ae i(kx#$t ) d 2 #(x) dx 2 E = p2 2m = E#(x) (spatial part of the wavefunction only) A slight generalisation takes into account the possibility that the particle has energy by virtue of being in a potential V as well as kinetic energy : "h 2 2m d 2 #(x) dx 2 + V#(x) = E#(x) Time-independent Schrödinger equation. 29 Time Dependence Going back to the wavefunction describing free particles of fixed momentum : We also expect particles described by this wavefunction to satisfy the second De Broglie relation : This suggests the equation : Solutions to this are simply : "(x,t) = Ae i(kx#$t ) ih d"(t) dt E = h" = E"(t) "(t) = Ae #iet / h (temporal wavefunction only) 30

16 Then the product wavefunction : satisfies : Time Dependence "(x,t) ="(x) #"(t) "h 2 2m d 2 # d# + V# = ih 2 dx dt Schrödinger equation. For the energy states we have been considering, the time-dependent factor e "iet / h does not change "(x) 2, so it was safe to treat them in a time-independent way. 31 The Hydrogen Atom (advanced) Degeneracy : Hidden in the energy level diagram for hydrogen is a high level of degeneracy : different quantum states that happen to have the same energy. For the hydrogen atom (in fact, for any quantum mechanical system with a central potential), each energy level has a well defined angular momentum. Quantum Numbers Principal Total Angular Momentum z-component of Angular Momentum n = 1 l = 0 n = 2 l = 0 l = 1 m = "1,0,+1 n = 3 l = 0 l = 1 l = 2 m = "1,0,+1 m = "2,"1,0,+1,+2 Angular Momentum = l(l + 1) h 32

17 Complex Atoms (advanced) Two more key principles are needed to understand atoms containing >1 electron : Pauli Exclusion Principle : no two fermions (e.g. electrons) can be in the same state. Electrons have half a unit of spin or intrinsic angular momentum : s = 1 2 " 3 2 h ; m s = # 1 2, different orientations : "up" & "down" This means that 2 electrons can occupy states identified by unique combinations of quantum numbers n, l and m. This material will be revisited in the lecture on the structure of matter 33 Complex Atoms (advanced) Energy-level diagram for a complex atom : Degeneracy has been broken (lifted). 34

Semiconductor Physics and Devices

Semiconductor Physics and Devices Introduction to Quantum Mechanics In order to understand the current-voltage characteristics, we need some knowledge of electron behavior in semiconductor when the electron is subjected to various potential