Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such that the eigevalues of the operator represets all the possible results that could be obtaied if the associated physical observable were to be measured. The eigestates of the operator are the states of the system for which the associated eigevalue would be, with 100% certaity, the measured result, if the observable were measured. If the system were i ay other state the the possible outcomes of the measuremet caot be predicted precisely differet possible results could be obtaied, each oe beig a eigevalue of the associated observable, but oly the probabilities ca be determied of these possible results. This physically observed state-of-affairs is reflected i the mathematical structure of quatum mechaics i that the eigestates of the observable form a complete set of basis states for the state space of the system, ad the compoets of the state of the system with respect to this set of basis states gives the probability amplitudes, ad by squarig, the probabilities of the various outcomes beig observed. Give the probabilistic ature of the measuremet outcomes, familiar tools used to aalyze statistical data such as the mea value ad stadard deviatio play a atural role i characterizig the results of such measuremets. The expectatio or mea value of a observable is a cocept take more or less directly from the classical theory of probability ad statistics. Roughly speakig, it represets the (probability-weighted) average value of the results of may repetitios of the measuremet of a observable, with each measuremet carried out o oe of very may idetically prepared copies of the same system. The stadard deviatio of these results is the a measure of the spread of the results aroud the mea value ad is kow, i a quatum mechaical cotext, as the ucertaity. 1.1 Observables with Discrete Values The probability iterpretatio of quatum mechaics plays a cetral, i fact a defiig, role i quatum mechaics, but the precise meaig of this probability iterpretatio has as yet ot bee fully spelt out. However, for the purposes of defiig the expectatio value ad the ucertaity it is ecessary to look a little more closely at how this probability is
Chapter 1 Probability, Expectatio Value ad Ucertaity 181 operatioally defied. The way this is doe depeds o whether the observable has a discrete or cotiuous set of eigevalues, so each case is discussed separately. We begi here with the discrete case. 1.1.1 Probability If A is a observable for a system with a discrete set of values {a 1, a,...}, the this observable is represeted by a Hermitea operator  that has these discrete values as its eigevalues, ad associated eigestates { a, = 1,, 3,...} satisfyig the eigevalue equatio  a = a a. These eigestates form a complete, orthoormal set so that ay state ψ of the system ca be writte where, provided ψ ψ = 1, the ψ = a a ψ a ψ = probability of obtaiig the result a o measurig Â. To see what this meas i a operatioal sese, we ca suppose that we prepare N idetical copies of the system all i exactly the same fashio so that each copy is presumably i the same state ψ. This set of idetical copies is sometimes called a esemble. If we the perform a measuremet of  for each member of the esemble, i.e. for each copy, we fid that we get radomly varyig results e.g. Copy: 1 3 4 5 6 7 8... N Result: a 5 a 3 a 1 a 9 a 3 a 8 a 5 a 9... a 7 Now cout up the umber of times we get the result a 1 call this N 1, ad similarly for the umber of times N we get the result a ad so o. The N N = fractio of times the result a is obtaied. (1.1) What we are therefore sayig is that if the experimet is performed o a icreasigly large umber of idetical copies, i.e. as N, the we fid that N lim N N = a ψ. (1.) This is the so-called frequecy iterpretatio of probability i quatum mechaics, i.e. the probability predicted by quatum mechaics gives us the frequecy with which a outcome will occur i the limit of a very large umber of repetitios of a experimet. There is aother iterpretatio of this probability, more aki to bettig odds i the sese that i applicatio, it is usually the case that the experimet is a oe-off affair, i.e. it is ot possible to repeat the experimet may times over uder idetical coditios. It the
Chapter 1 Probability, Expectatio Value ad Ucertaity 18 makes more sese to try to uderstad this probability as a measure of the liklihood of a particular result beig actually observed, i the same way that bettig odds for a horse to wi a race is a measure of the cofidece there is i the horse actually wiig. After all, a race is oly ru oce, so while it is possible to imagie the race beig ru may times over uder idetical coditios, ad that the odds would provide a measure of how may of these imagied races a particular horse would wi, there appears to be some doubt of the practical value of this lie of thikig. Nevertheless, the probability as frequecy iterpretatio of quatum probabilities is the iterpretatio that is still most commoly to be foud i quatum mechaics. 1.1. Expectatio Value We ca ow use this result to arrive at a expressio for the average or mea value of all these results. If the measuremet is repeated N times, the average will be N 1 N a 1 + N N a N +... = N a. (1.3) Now take the limit of this result for N, ad call the result  : N  = lim N N a = a ψ a. (1.4) It is a simple matter to use the properties of the eigestates of  to reduce this expressio to a compact form:  = ψ a a ψ a = ψ  a a ψ { } = ψ  a a ψ = ψ  ψ (1.5) where the completeess relatio for the eigestates of  has bee used. This result ca be readily geeralized to calculate the expectatio values of ay fuctio of the observable Â. Thus, i geeral, we have that f (Â) = a ψ f (a ) = ψ f (Â) ψ (1.6) a result that will fid immediate use below whe the ucertaity of a observable will be cosidered. While the expectatio value has bee defied i terms of the outcomes of measuremets of a observable, the same cocept of a expectatio value is applied to o-hermitea operators which do ot represet observable properties of a system. Thus if  is ot Hermitea ad hece ot a observable we caot make use of the idea of a collectio of results of experimets over which a average is take to arrive at a expectatio value. Nevertheless, we cotiue to make use of the otio of the expectatio value  = ψ  ψ eve whe  is ot a observable.
Chapter 1 Probability, Expectatio Value ad Ucertaity 183 1.1.3 Ucertaity The ucertaity of the observable A is a measure of the spread of results aroud the mea Â. It is defied i the usual way, that is the differece betwee each measured result ad the mea is calculated, i.e. a Â, the the average take of the square of all these differeces. I the limit of N repetitios of the same experimet we arrive at a expressio for the ucertaity ( A) : ( A) N = lim N N (a  ) = a ψ (a  ). (1.7) As was the case for the expectatio value, this ca be writte i a more compact form: ( A) = ψ a a ψ (a  ) = ψ a a ψ (a a  +  ) = ψ a a ψ a  ψ a a ψ a +  ψ a a ψ Usig f (Â) a = f (a ) a the gives { } ( A) = ψ  a a ψ  ψ  a a ψ +  ψ a a ψ = ψ  { } a a ψ  ψ  { } a a ψ +  ψ ψ Assumig that the state ψ is ormalized to uity, ad makig further use of the completeess relatio for the basis states a this becomes ( A) = ψ  ψ  +  =  Â. (1.8) We ca get aother useful expressio for the ucertaity if we proceed i a slightly differet fashio: ( Â) = ψ a a ψ (a  ) Now usig f (Â) a = f (a ) a the gives = ( Â) = ψ a (a  ) a ψ ψ (  ) a a ψ { = ψ (  ) = ψ (  ) ψ } a a ψ = (  ). (1.9)
Chapter 1 Probability, Expectatio Value ad Ucertaity 184 Thus we have, for the ucertaity Â, the two results ( Â) = (  ) =  Â. (1.10) Just as was the case for the expectatio value, we ca also make use of this expressio to formally calculate the ucertaity of some operator  eve if it is ot a observable, though the fact that the results caot be iterpreted as the stadard deviatio of a set of actually observed results meas that the physical iterpretatio of the ucertaity so obtaied is ot particularly clear. Ex 1.1 For the O io (see pp 88, 9) calculate the expectatio value ad stadard deviatio of the positio of the electro if the io is prepared i the state ψ = α + a + β a. I this case, the positio operator is give by so that ˆx = a + a +a a a a ˆx ψ = aα + a aβ a ad hece the expectatio value of the positio of the electro is which ca be recogized as beig just ψ ˆx ψ = ˆx = a ( α β ) ˆx = (+a) probability α of measurig electro positio to be +a + ( a) probability β of measurig electro positio to be a. I particular, if the probability of the electro beig foud o either oxyge atom is equal, that is α = β = 1, the the expectatio value of the positio of the electro is ˆx = 0. The ucertaity i the positio of the electro is give by ( ˆx) = ˆx ˆx. We already have ˆx, ad to calculate ˆx it is sufficiet to make use of the expressio above for ˆx ψ to give ˆx ψ = ˆx [ aα + a aβ a ] = a [ α + a + β a ] = a ψ so that ψ ˆx ψ = ˆx = a. Cosequetly, we fid that ( ˆx) = a a ( α β ) = a [ 1 ( α β ) ] = a ( 1 α + β )( 1 + α β )
Chapter 1 Probability, Expectatio Value ad Ucertaity 185 Usig ψ ψ = α + β = 1 this becomes from which we fid that ( ˆx) = 4a α β ˆx = a αβ. This result is particularly illumiatig i the case of α = β = 1 where we fid that ˆx = a. This is the symmetrical situatio i which the mea positio of the electro is ˆx = 0, ad the ucertaity is exactly equal to the displacemet of the two oxyge atoms o either side of this mea. Ex 1. As a example of a istace i which it is useful to deal with the expectatio values of o-hermitea operators, we ca tur to the system cosistig of idetical photos i a sigle mode cavity (see pp 95, 1, 174). If the cavity field is prepared i the umber state, a eigestate of the umber operator ˆN defied i Eq. (11.54), the we immediately have ˆN = ad ˆN = 0. However, if we cosider a state such as ψ = ( + + 1 )/ we fid that while ˆN = 1 ( + + 1 ) ˆN( + + 1 ) ( ˆN) = ˆN ˆN = 1 ( + + 1 )( + ( + 1) + 1 ) = + 1. = 1 ( + + 1 ) ˆN ˆN( + + 1 ) ( + 1 ) = 1 ( + ( + 1) + 1 )( + ( + 1) + 1 ) ( + 1 ) = 1 ( + ( + 1) ) ( + 1 ) = 1 4. Not uexpectedly,we fid that the ucertaity i the photo umber is ow ozero. We ca also evaluate the expectatio value of the aihilatio operator â, Eq. (9.56) for the system i each of the above states. Thus, for the umber state we have â = â = 1 = 0 while for the state ψ = ( + + 1 )/ we fid that â = 1. It is a iterestig exercise to try to give some kid of meaig to these expectatio values for â, particularly sice the expectatio value of a o-hermitea operator does ot, i geeral, represet the average of a collectio of results obtaied whe measurig some observable. The meaig is provided via the demostratio i Sectio 11.4.4 that â is directly related to the electric field stregth iside the cavity. From that demostratio, it ca be see that the real part of â is to be uderstood as beig proportioal to the average stregth of the electric (or magetic) field iside the cavity, i.e. Re â Ê where Ê is
Chapter 1 Probability, Expectatio Value ad Ucertaity 186 the electric field operator. We ca the iterpret the expressios obtaied above for the expectatio value of the field as sayig little more tha that for the field i the umber state, the electric field has a average value of zero i.e. if the photo umber is kow precisely, the the field averages out to zero. This result ca be give a classical iterpretatio, i.e. that the electric field has a completely radom phase so that the time averaged field will be zero. I cotrast, if the field is i the state ψ = ( + + 1 )/ for which the ucertaity i the photo umber is ˆN = 1, we fid that the average electric field is o loger zero. Classically this ca 4 be uderstood as the field havig a more well-defied phase. This suggests that there is a balacig act of sorts betwee the radomess of the phase of the field, i.e. how well this phase is defied, ad the ucertaity i the umber of photos there are i the field: makig the photo umber less ucertai results i icreased radomess, or ucertaity, i the phase of the field. This is ot a good example to explore this issue ay further, i part because the cocept of a phase observable is a cotroversial oe, but it does give a taste of a kid of behaviour to foud throughout quatum mechaics, amely that if oe observable is kow with precisio the typically some other observable is less precisely kow. This relatioship is embodied i the Heiseberg Ucertaity Relatio which we will be lookig at later. For the preset we will be cosiderig the ideas of expectatio value ad ucertaity i the case of a observable with cotiuous eigevalues. 1. Observables with Cotiuous Values Much of what has bee said i the precedig Sectio carries through with relatively mior chages i the case i which the observable uder cosideratio has a cotiuous eigevalue spectrum. However, rather tha discussig the geeral case, we will cosider oly the particular istace of the measuremet of the positio represeted by the positio operator ˆx of a particle. 1..1 Probability Recall from Sectio 11.3., p164, that i order to measure the positio a observable with a cotiuous rage of possible values it was ecessary to suppose that the measurig apparatus had a fiite resolutio δx i which case we divide the cotiuous rage of values of x ito itervals of legth δx, so that the th segmet is the iterval (( 1)δx, δx). If we let x be the poit i the middle of the th iterval, i.e. x = ( 1 )δx, we the say that the particle is i the state x if the measurig apparatus idicates that the positio of the particle is i the th segmet. The observable beig measured is ow represeted by the Hermitea operator ˆx δx with discrete eigevalues {x ; = 0, ±1, ±...} ad associated eigevectors { x ; = 0, ±1, ±,...} such that ˆx δx x = x x.
Chapter 1 Probability, Expectatio Value ad Ucertaity 187 We ca ow imagie settig up, as i the discrete case discussed above, a esemble of N idetical systems all prepared i the state ψ, ad the positio of the particle measured with a resolutio δx. We the take ote of the umber of times N i which the particle was observed to lie i the th iterval ad from this data costruct a histogram of the frequecy N /N of the results for each iterval. A example of such a set of results is illustrated i Fig. 1.1. As was discussed earlier, the claim is ow made that for N large, the frequecy with which the particle 50 45 40 35 30 5 0 15 10 5 0-5 N Nδx -4-3 - -1 0 1 P(x) = x ψ Figure 1.1: Histogram of the frequecies of detectios i each iterval (( 1)δx, δx), = 0, ±1, ±,... for the measuremet of the positio of a particle for N idetical copies all prepared i the state ψ ad usig apparatus with resolutio δx. Also plotted is the probability distributio x ψ expected i the limit of N ad δx 0. is observed to lie i the th iterval (( 1)δx, δx), that is N /N, should approximate to the probability predicted by quatum mechaics, i.e. 3 4 5 6 N N x ψ δx (1.11) or, i other words N Nδx x ψ. (1.1) Thus, a histogram of the frequecy N /N ormalized per uit δx should approximate to the idealized result x ψ expected i the limit of N. This is also illustrated i Fig. 1.1. I the limit of the resolutio δx 0, the tops of the histograms would approach the smooth curve i Fig. 1.1. All this amouts to sayig that ( N ) lim lim = x ψ. (1.13) δx 0 N Nδx The geeralizatio of this result to other observables with cotiuous eigevalues is essetially alog the same lies as preseted above for the positio observable. 1.. Expectatio Values We ca ow calculate the expectatio value of a observable with cotiuous eigevalues by makig use of the limitig procedure itroduced above. Rather tha cofiig our attetio to the particular case of particle positio, we will work through the derivatio of the expressio for the expectatio value of a arbitrary observable with a cotiuous eigevalue spectrum. Thus, suppose we are measurig a observable  with a cotiuous rage of eigevalues α 1 < a < α. We do this, as idicated i the precedig Sectio by supposig that the rage of possible values is divided up ito small segmets of legth δa. Suppose the measuremet is repeated N times o N idetical copies of the same system, all prepared i the same state ψ. I geeral, the same result will ot be obtaied every
Chapter 1 Probability, Expectatio Value ad Ucertaity 188 time, so suppose that o N occasios, a result a was obtaied which lay i the th iterval (( 1)δa, δa). The average of all these results will the be N 1 N a 1 + N N a +.... (1.14) If we ow take the limit of this result for N, the we ca replace the ratios N /N by a ψ δa to give a 1 ψ a 1 δa + a ψ a δa + a 3 ψ a 3 δa +... = a ψ a δa. (1.15) Fially, if we let δa 0 the sum becomes a itegral. Callig the result Â, we get:  = α α 1 a ψ ada. (1.16) It is a simple matter to use the properties of the eigestates of  to reduce this expressio to a compact form:  = α α 1 α ψ a a a ψ da = ψ  a a ψ da α 1 { α = ψ  α 1 } a a da ψ = ψ  ψ (1.17) where the completeess relatio for the eigestates of  has bee used. This is the same compact expressio that was obtaied i the discrete case earlier. Oce agai, as i the discrete case, if  is ot Hermitea ad hece ot a observable we caot make use of the idea of a collectio of results of experimets over which a average is take to arrive at a expectatio value. Nevertheless, we cotiue to make use of the otio of the expectatio value as defied by  = ψ  ψ eve whe  is ot a observable. I the particular case i which the observable is the positio of a particle, the the expectatio value of positio will be give by ˆx = + x ψ x dx = + ψ(x) x dx (1.18) 1..3 Ucertaity 1.3 The Heiseberg Ucertaity Relatio 1.4 Compatible ad Icompatible Observables 1.4.1 Degeeracy