Chapter 4 Time{Independent Schr odinger Equation

1 Chapter 4 Time Independent Schr Stationary StatesWe consider again the time dependent Schr odinger Equation (Prop. )i~ t (t,x) =( ~22m +V(x)) (t,x) =H (t,x),( )where the potential in the Hamiltonian is assumed to be time independentV=V(x) .We calculate the solutions of this Equation by using themethod of separation of variables, we make the following ansatz for the solution (t,x): (t,x) = (x)f(t)( )and insert it into the time dependent Schr odinger Equation , Eq. ( ),i~ (x) f(t) t= ~22m 2 (x) x2f(t) +V(x) (x)f(t)| 1 (x)f(t)i~1f(t)df(t)dt= ~22m1 (x)d2 (x)dx2+V(x).( )Since now the left hand side in Eq. ( ) is only dependent ontand the right handside only onx, both sides must be equal to a constant, which we will callE, and we canthus solve each side independently. The left side yieldsi~1f(t)df(t)dt=E dff= i~E dt ln(f) = i~E t+ const. f= iE t/~.( )The constant in Eq. ( ) will later on be absorbed into (x).6970 Chapter 4. TIME INDEPENDENT SCHR odinger EQUATIONThen multiplying the right side of Eq.

2 ( ) with (x) we get ~22m1 (x)d2 (x)dx2+V(x) =E ~22md2 (x)dx2+V(x) (x) H (x)=E (x).( )The operators on the left express the HamiltonianHacting on (x), which representsthetime independent Schr odinger (Time-independent Schr odinger Equation )H (x) =E (x)whereH= ~22m +V(x)is the HamiltonianDefinition state is calledstationary, if it is represented by the wave function (t,x) = (x)e iE t/~.For such states the probability density is time independent| (t,x)|2= (x) (x)eiE t/~e iE t/~ 1=| (x)|2.( )The expectation values of observablesA(X,P) are time independent as well A(X,P) = dx (x)eiE t/~A(x, i~ x) (x)e iE t/~= dx (x)A(x, i~ x) (x).( )Remark I:As a consequence, the eigenvalues of the Hamiltonian, which are thepossible energy levels of the system, are clearly time see it, just takeH(X,P) instead ofA(X,P) in Eq. ( ) and use the time-independent Schr odinger Equation (Theorem ) H(X,P) = dx (x)H (x) = dx (x)E (x) =E dx (x) (x) <.

3 ( ) STATIONARY STATES71 Remark II:The normalization of the wavefunction will restrict the possible valuesof the constantE, the energy of the system, in the Schr odinger more interesting features about stationary states and the corresponding energieswill be formulated here in the form of two lemmata, whose proofs we will leave as normalizable solutions (x)of the Schr odingerequation the energyEmust be real,E (x)of the time-independent Schr odingerequation can always be chosen to be operatorPacting on a functionf(x)changes the sign of its argument:Pf(x) =f( x).We conclude that even and odd functions are eigenfunctions of the parity operatorP even= + evenP odd= odd,( )which we will use in the following theorem that will be helpful later a symmetric potentialV(x) =V( x)a basis of states can be chosen,that consists entirely of even and odd functions. even(x) = (x) + ( x) odd(x) = (x) ( x)The proof for this theorem will be left as an exercise 4.

4 TIME INDEPENDENT SCHR odinger Schr odinger Equation as eigenvalue EquationA subject concerning the time-independent Schr odinger Equation we have not yet touchedis its interpretation as an eigenvalue Equation . Clearly, from its form we see that stationarystates| are eigenvectors/eigenfunctions of the HamiltonianHwith eigenvaluesEH| =E| .( )It implies the exact determination of the energyE. A stationary state has a preciselydefined energy. Calculating the expectation value of the Hamiltonian for a stationarysystem just gives H = |H| = |E| =E | =E .( )Consequently, there is no energy uncertainty Efor these states E= H= H2 H 2= E2 E2= 0.( )Generally eigenvalue equations for linear operators take the formA| =a| ,( )whereais an eigenvalue of the linear operatorAwith corresponding eigenvector| .For hermitian operators there exist important statements about their eigenvalues eigenvalues of hermitian operators are real and the eigenvectors corre-sponding to different eigenvalues are proof is easy and again left as an exercise.

5 The above theorem is vitally importantfor the spectrum{En}of the Hamiltonian, which is thereby guaranteed to be realH| n =En| n .( )Using our notation| n |n the orthogonality and completeness relations (re-member equations ( ) and ( )) can be written as n|m = nm n|n n|=1.( ) EXPANSION INTO STATIONARY Expansion into Stationary StatesUsing the spectral theorem (Theorem ) we can then expand a given state into a com-plete orthonormal system of energy eigenstates|n exactly as outlined in Section | = ncn|n cn= n| .( )By inserting a continous CONS of position eigenstates (Eq. ( )) into the transitionamplidute the expansion coefficientscncan be rewritten ascn= n| = dx n|x x| = dx n(x) (x).( )We can now extend the expansion from the time independent case to the time depen-dent one. We just remember the time dependent Schr odinger equationi~ t (t,x) =H (t,x),( )with a particular solution n(t,x) = n(x)e iEnt/~.( )The general solution is then a superposition of particular solutions (t,x) = ncn n(x)e iEnt/~.

6 ( )The expansion coefficients can easily be computed by settingt= 0 and taking the scalarproduct with m(x) dx m(x) (0,x) = dx m(x) ncn n(x)e iEn0/~ 1 m| (t= 0) = ncn dx m(x) n(x) the expansion coefficients are given bycn= n| (t= 0) .( )Physical interpretation of the expansion coefficients:Let s consider an observableAwith eigenstates nand eigenvaluesanA| n =an| n .( )If a system is in an eigenstate of this observable the expectation value (in this state) isequal to the corresponding eigenvalue A = n|A| n =an n| n =an.( )74 Chapter 4. TIME INDEPENDENT SCHR odinger EQUATIONThus a measurement of the observable always produces the resultanwhich impliesthat the uncertainty of the observable vanishes for this state A= 0. Furthermore themeasurement leaves the state unchanged, the system remains in the eigenstate| n A | n .( )If the system, however, is in a general state| , which is a superposition of eigenstates,the expectation value is given by the sum of all eigenvalues, weighted with the modulussquared of the expansion coefficients A = |A| = n m cm m|A|cn n = n mc mcnan m| n mn= n|cn|2an.

7 ( )The expansion coefficientscn= n| can thus be regarded as a probability am-plitude for the transition from a state to an eigenstate nwhen the correspondingobservable is measured. The actual transition probability is given by its modulus squared|cn|2 theprobability for measuring the resultan which also obeys n|cn|2= 1.( )So a measurement of an observable in a general state changes the state to one of theeigenstates of the observable. This process is often called thereductionorcollaps of thewave function| A | n .( ) Infinite Potential WellOur goal in the next sections is to calculate the energy eigenvalues and eigenfunctions forseveral Hamiltonians, for several potentials. Let us begin with the infinite potentialwell, represented by the potentialV(x), as illustrated in Fig. , such thatV(x) ={0 forx [ 0, L] else( )This means that the quantum object is limited to a certain region betweenx= 0 andx=Lwhere it moves freely but cannot ever leave.}

8 Thus mathematically we have (x) = 0 forx / [ 0, L].( ) INFINITE POTENTIAL WELL75 Figure : Infinite potential well: The potential is infinite outside the interval [ 0, L],inside it vanishes. Therefore the only physically allowed region for a particle is inside , for the wave function to be continuous we have to require that it vanishesat the boundaries (0) = (L) = 0.( )The only region were particles are allowed is inside the well, where they behave likefree particles, they are not exposed to a potential. Therefore we need to solve the free(time-independent) Schr odinger Equation with the boundary conditions from Eq. ( ) ~22md2dx2 (x) =E (x).( )With the abbreviationk2=2mE~2,k= 2mE~( )the free Schr odinger Equation takes the following formd2dx2 (x) = k2 (x),( )where the general solution is well known, and given by (x) =asin(kx) +bcos (kx).( )Hereaandbare some constants that are yet to be determined by the boundary conditions,starting with (0) = 00 = (0) =asin(0) 0+bcos(0) b= 0.

9 ( )76 Chapter 4. TIME INDEPENDENT SCHR odinger EQUATIONE xploiting the second boundary condition (L) = 0 , leads to discrete values ofk0 = (L) =asin(kL) kL=n k=n L,( )wheren= 1,2,3,..can be any natural number. Inserting our result into Eq. ( )and solving it with respect toEwe see that theenergy is quantized. Labeling the severalenergy levels bynwe findEn=n2 2~22mL2.( )Finally, the value of the constantafollows from the normalization of the wave functionL 0dx| |2= 1 |a|2L 0dxsin2(n Lx) = 1 |a|2=2L.( )Thus the bound states of the infinite potential well, which form a CONS, are thengiven by n(x) = 2 Lsin(n Lx).( )Forn= 1 we get the ground state energy and wave functionE1, 1of the infinitepotential well, the higher states withn >1 are calledexcited Finite Potential WellWe now study a similar problem as in Section , but with the change that the potentialwalls are no longer infinitely high. Classically, a particle is trapped within the box, if itsenergy is lower than the height of the walls, , it has zero probability of being foundoutside the box.

10 We will see here that, quantum mechanically, the situation is time-independent Schr odinger Equation is again our starting point where we nowinsert the following potentialV(x) into our HamiltonianV(x) ={ V0for|x| L0for|x|> L( )For the possible energy rangeE > V0we consider separately the two energy regions, V0< E <0 for thebound statesandE >0 for thescattered states. We also split thewholex-range into the three regions I, II, and III, where we solve the equations FINITE POTENTIAL Bound StatesRegion I:x < L, V(x) = 0 Here we have again the free Schr odinger Equation ~22md2dx2 (x) =E (x),( )which we rewrite by substituting =1~ 2mE, where >0 becauseE <0,d2dx2 (x) = 2 (x).( )We already know that the general solution of Eq. ( ) is given by (x) =Ae x+B e x,( )whereAandBare constants, yet to be determined. Since we are in the region wherex < L <0 the exponent of the first term would ever increase forx . In order tokeep the wave function normalizable we must demand that the constantAbe identicallyzero, and we get as solution for region I (x) =B e x.}