* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download REVIEW OF WAVE MECHANICS
Dirac equation wikipedia , lookup
Casimir effect wikipedia , lookup
Schrödinger equation wikipedia , lookup
Quantum group wikipedia , lookup
Identical particles wikipedia , lookup
Tight binding wikipedia , lookup
Second quantization wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum teleportation wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Quantum entanglement wikipedia , lookup
Renormalization wikipedia , lookup
Double-slit experiment wikipedia , lookup
Bell's theorem wikipedia , lookup
Scalar field theory wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Hydrogen atom wikipedia , lookup
Coupled cluster wikipedia , lookup
Path integral formulation wikipedia , lookup
Renormalization group wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Density matrix wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Coherent states wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Particle in a box wikipedia , lookup
Probability amplitude wikipedia , lookup
Canonical quantization wikipedia , lookup
Wave function wikipedia , lookup
Matter wave wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum state wikipedia , lookup
Hidden variable theory wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran EXPECTATION VALUE AND UNCERTAINTY 2.1 Introduction You may be aware of the special role eigenvalue equations play in quantum mechanics. Many of the important equations you have seen have been in the form of Q n q n n , where Q is an operator (e.g. the Hamiltonian on the left hand side of Schrodinger’s equations), and qn and n respectively are the eigenvalues and eigenfunctions of this operator labelled by quantum number(s) n(lm...). Then a measurement of the quantity represented by the operator Q on the system whose state is represented by n will definitely result in the answer qn. What happens if the wave function of the system is not an eigenfunction of the operator Q ? In this case the answer to the measurement will be one of Q ’s eigenvalues, but it is not certain which one it will be! There is however a well tested mathematical procedure for calculating the probability of each possible answer occurring as we shall see later in the course. Thus in quantum mechanics we typically calculate the probabilities of results of measurements. You are already familiar with this idea through the interpretation of the wave function, where * gives the probability of finding the particle at a given position. This restriction to probability rather than certainty extends to all physical measurements, not just position. The only exceptions are when the wave function representing the state of a system happens to be an eigenfunction of the operator representing the physical quantity, and in these cases the outcome of the measurement will be known with probability one. It is convenient to characterise the distribution of possible results of the measurements. The first useful statistical quantity is the average value of the result obtained from a large number of repeated measurements on identical states of the system. In the quantum mechanics this is called the expectation . value of the operator, and is denoted by Q The second statistical characteristic is the standard deviation of the measured values. It is called the uncertainty in the result q in quantum mechanics and is denoted by q . This is the quantity that is used in Heisenberg’s uncertainty relations. 1 3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran 2.2 Expectation Values Expectation value for position measurement: Since the probability of find the particle between x and x+dx is given by u*(x)u(x).dx, where the spatial wave function has been normalised, x u * ( x ). x. u ( x ). dx This way of writing the product of u*.u.x = u*.x.u is consistent with the general formula described below. Similarly for any function of position, such as potential energy, we have V u * ( x ).V ( x ). u ( x ). dx To find a suitable expression for other operators which are not simple functions of position but may involve differentiation, we take guidance from the TISE: ( x ) V ( x ). u ( x ) E . u ( x ) , Tu 2 d 2 where T is the kinetic energy operator. 2 dx 2 Multiplying by u*(x) and integrating over all space we find ( x ). dx V E . u * (x ). Tu Now classically the total energy E = T + V, the sum of kinetic and potential energies. We certainly expect to find the same result from the averages taken over a large number of measurements on identical systems, E T V . We thus identify the expectation value of the kinetic energy operator to be given by ( x ). dx . T u * ( x ). Tu 2 3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran We now suppose that there is nothing unique about the kinetic energy operator, so that the formula for all expectation values is ( x ). dx . Q u * ( x ). Qu N.B. The operator Q operates on the wave function(s) to its right! Whilst we have looked at the one dimensional case here, generalisation to higher dimensions is obvious. as the average of a large It is important to remember the definition of Q number of measurements taken on identical systems represented by the wave function u(x). This is not the same as the most likely result of a single measurement; statistically the mean is not necessarily equal to the mode! 2.3 Uncertainty The standard deviation in a set of results is a measure of how uncertain we would be about the value of a single measurement. In statistics it is derived as the root-mean-square deviation from the average value, q (q q ) 2 q 2 q 2 . This same expression is adopted in quantum mechanics, where the expectation value formula derived above is used to calculated the uncertainty in the result of a measurement of a physical quantity represented by the operator Q , q Q 2 Q 2 . Using these definitions of uncertainty, Heisenberg’s Uncertainty Principle for position and momentum reads x. p x 1 . 2 In some texts the right-hand-side may be quoted differently, although it will always involve Planck’s constant. This means that they are using slightly different definitions of uncertainty to ours and is not a cause for concern. The principle is the same, that there is limit to how well conjugate variables such as the x-component of position and the x-component of momentum can be known simultaneously; more on this later. 3 3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran WORKSHOP QUESTIONS Hand your solutions to the following questions to Dr. Mulheran at the start of the second workshop in week 3. Some of your solutions will be marked as part of the continuous assessment of this course which contributes 20% of the overall module grade. Your solutions must be well presented; untidy work will be penalised. 2.1 The normalised ground state wave function for the one dimensional harmonic oscillator is 2 u0 ( x ) x , exp 2 where as usual is the particle’s mass and is the classical angular frequency of the oscillator. (a) Calculate the expectation values for position and momentum, x and Px , with this wave function and comment on your results. [2 marks] Hint: think about the symmetry of the functions you must integrate! (b) Calculate the expectation values x 2 and P 2 . x [3 marks] Hint: the following standard integral will be of help: 1/ 4 x 2 exp(x 2 )dx (c) (d) 2.2 . 2 3/ 2 Hence calculate T and V , where T is the kinetic energy operator, and show that the sum of these equals the total energy of the ground state as required. [3 marks] Using the results from (a) and (b), now calculate the uncertainties in the particle’s position and momentum. Is Heisenberg’s Uncertainty Principle obeyed? [2 marks] A bead which slides freely around a wire hoop (as in question 1.2) has the normalised wave function u( ) 1 sin( ) . The angular momentum operator for this situation (i.e. one with cylindrical symmetry) is L z i . 4 3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran (a) (b) (c) Express u() as a linear superposition of the eigenfunctions of L̂ z . [2 marks] 2 Calculate Lz , Lz and l z for the bead and comment on your results. [3 marks] Hint: the integrals used in the expectation values run from 0 to 2 in this case. Teacher’s Pet question: Bonus marks are available for anyone able to show why the angular momentum operator for cylindrical systems is as given above. See Dr. Mulheran for hints! 5 3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran WORKSHOP SOLUTIONS 2.1 (a) u(x ) A exp( x 2 ) x A exp( x 2 ). x. A exp( x 2 ) dx 0 since the integrand is an odd function of x and we integrate over all x. Similarly Px A exp( A exp( d x 2 ). i . A exp( x 2 ). dx dx x 2 ).i 2 xA exp( x 2 ). dx 0 These results are obviously true, since the wave function is evenly spread about the origin, and the particle is equally likely to be found moving the right as it is moving to the left. (b) A exp( x 2 x 2 ). x 2 . A exp( x 2 ) dx 8 3 A 5/ 2 3/ 2 . 2 32 3 3 2 2 u’=-2ax.u, u’’=-2a.u+4a2x2u, Px2 2 ( 2 4 2 x 2 ) 4 2 2 . . 2 4 2 2 1 T Px2 2 4 1 V 2 x 2 2 4 Clearly T V which is the ground state energy. 2 x x 2 x 2 0, 2 2 (c) (d) p x Px2 Px 2 0. 2 Thus x. p x , so Heisenberg’s Uncertainty Principle is obeyed. In fact 2 the ground state wave function of the harmonic oscillator, which is a Gaussian function, is the minimum uncertainty wave packet possible, and the equality in the HUP is not normally found even for ground state wave functions in general systems. 6 3/PH/SB Quantum Theory - Week 2 - Dr. PA Mulheran 2.2 (a) The eigenfunctions of L̂ z are u (b) 1 2i 2 1 2 exp im with integer m. Thus exp i exp i . 1 2 d i 2 L z sin( ). i sin( )d 0 sin( ).cos( )d 0 0 d 2 2 2 d 1 2 2 2 2 Lz sin( ). sin( )d sin ( )d 2 2 0 d 0 lz Using Euler, we see that the wave function is a linear superposition of two states with angular momentum . Thus the average of a large number of measurements is zero, and the uncertainty is . 7