C CS Phys C191 Quantum Mechanics in a Nutshell Fall 2009 10 06 07 Lecture 12 In this lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to this course Topics in this set of notes Planck Einstein postulates operators representations uncertainty principle Planck Einstein Relation E h This is the equation relating energy to frequency It was the earliest equation of quantum mechanics implying that energy comes in multiples quanta of a fundamental constant h It is written as either E h or E h where h h 2 is linear frequency and is angular frequency The fundamental constant h is called Planck s constant and is equal to 6 62608 10 34 Js h 1 05457 10 34 Js or 1 05457 10 27 erg s This relation was first proposed by Planck in 1900 to explain the properties of black body radiation The interpretation was that matter energy levels are quantized At the time this appeared compatible with the notion that matter is composed of particles that oscillate The discovery that the energy of electrons in atoms is given by discrete levels also fitted well with the Planck relation In 1905 Einstein proposed that the same equation should hold also for photons in his explanation of the photoelectric effect The light incident on a metal plate gives rise to a current of electrons only when the frequency of the light is greater than a certain value This value is associated with the energy required to remove an electron from the metal the work function The electron is ejected only when the light energy matches the discrete electron binding energy Einstein s proposal that the light energy is quantized just like the electron energy was more radical at the time light quantization was harder for people to accept than quantization of energy levels of matter particles The word photon for these quantized packets of light energy came later given by G N Lewis of Lewis Hall 1 Fundamental physical postulates Why do quantum state evolve in time according to this particular operator and what is the meaning of this operator To answer this we have to look at quantum mechanics from a more physical perspective The physical basis of quantum mechanics rests on three fundamental postulates These are given below in the wording of K Gottfried and T M Yan Quantum Mechanics Fundamental Springer 2003 I States superposition The most complete description of the state of any physical system S at any time is provided by some vector v in the Hilbert space H appropriate to the system Every linear combination of such state vectors represents a possible physical state of S This last sentence is the superposition principle that we have been using from the very beginning Note the difference between a quantum and a classical description of a physical system A classical description is complete with specification of the positions and momenta of all particles each of which can be precisely measured at any time In contrast the quantum description is specified by the wave function that lives C CS Phys C191 Fall 2009 Lecture 12 1 in an abstract Hilbert space that has no direct connection to the physical world Classical mechanics is deterministic particle positions and momenta can be specified for all times using the classical equations of motion In contrast quantum mechanics provides a statistical prediction of the outcomes of all observables on the system as the wave function evolves Both descriptions are complete but they differ in the information that can be obtained The uncertainty principle fundamentally changes the relation between coordinates and momenta in quantum mechanics II Observables The physically meaningful entities of classical mechanics such as position q or x momentum p etc are represented by Hermitian operators Following Dirac we refer to these as observables We generalize these today to any physical meaningful entities i e including those observables that have no classical correspondence e g intrinsic spin III Probabilistic interpretation and Measurement A set of N replicas of a quantum system S described by a state when subjected to measurements for a physical observable A will yield in each measurement one of the eigenvalues a1 a2 of A and as N this eigenvalue will appear with probability P a1 P a2 where P ai ai 2 and ai is the eigenvector corresponding to the eigenvalue ai This is precisely the definition of probability in terms of specific outcomes in a sequence of identical tests on copies of S provided that P ai i ai 2 1 i This is automatically satisfied for states that are normalized to unity The expectation value of an observable A in an arbitrary state also looks like an average over a probability distribution hAi A ai ai ai i ai P ai i Note that if the state is an eigenstate of A then A aj where a j is the corresponding eigenvalue i e only a single term contributes We can generalize this procedure from projection onto eigenstates to projection onto an arbitrary state Thus the probability to find a quantum system S that is in state in another state is equal to P 2 This projection of the ket onto another state be it an eigenfunction of some operator ai a basis function for the Hilbert space vi or an arbitrary state is referred to as a probability amplitude since its square modulus is a probability Note that the probability amplitude is specified both by and the other state the latter specifies the representation of which realizes the quantum state in a measurable basis The probability amplitude is also referred to as the wave function in the specified representation C CS Phys C191 Fall 2009 Lecture 12 2 A single measurement of th observable A on a state in the basis representation of eigenstates of A will yield the value ai with probability P ai ai 2 This defines the measurement operator M i ai ai that acts on the state The normalized state after measurement is then easily seen to be equal to M i q Mi Mi For a measurement in the ai basis this is given by i q i Mi Mi where we have abbreviated ai i For example suppose we have the linear superposition 1 1 2 2 3 3 k k Making a single measurement of the observable A on will result in the outcome ai with probability P ai i 2 and the resulting state after the measurement is equal to i i i The measurement of the observable has collapsed the state to a single eigenstate i ai of A recall these constitute an orthonormal basis 2 Operators In an earlier lecture we defined the operator P ih which projects an arbitrary state onto the state i