Chapter Eight The Formal Structure of Quantum Mechanical Function Spaces

Transcription

1 Chapter Eight The Formal Structure of Quantum Mechanical Function Spaces Introduction In this chapter, e introduce the formal structure of quantum theory. This formal structure is based primarily on the concepts of linear vector spaces. Let's revie the formalism of representing the position of a particle in three-dimensional space using conventional 3-dimensional vectors. First e establish some arbitrary coordinate system, hich e designate here by the coordinates B, C, and D. We can then rite the vector V t in terms of unit vectors (i.e., in terms of predefined vectors hich are fixed in direction along the B, C, and D directions and hich are of unit length) according to the equation Vt œ B s3 C s4 D s 5 (.1) z R θ φ y x These unit vectors s3, s4, and s5 are specifically chosen to be perpendicular to each other and of unit length, so that s3 3 s œ ", and s3 4 s œs3 5 s œ!, etc. This means that if e take the dot product of one of the unit vectors (say s3 ) ith the vector Vt e obtain s3 V t œs3 3B 3 4C s s s 3 5D s s œ B (.2) Therefore, the component of the vector V t in a given direction can be ritten as the projection of the vector onto the corresponding unit vector. B œs3 V t C œs4 V t D œs5 Vt (.3) These unit vectors are said to span the space because any vector V t can be represented by some linear combination of these unit vectors. Although e could span three

2 Quantum Mechanics (Griffith) 2 dimensional space ith any three unit vectors hich are not co-linear, the choice of orthogonal unit vectors makes things somehat simpler. Let's change notation a bit so that e can rite things in a form more normal to quantum mechanics. We could rite the unit vectors in the form lbùß lcùß and ldù so that the vector V t is represented by the equation lvù œ BlBÙ ClCÙ DlDÙ (.4) If e anted to extend our notation to be valid for cases here there are more than three dimensions, e ould need to introduce ne coordinates and components in addition to BßCß and D. This is often accomplished by introducing a different notation for our unit vectors lbù œ l/ " Ù lcù œ l/ # Ù ldù œ l/ Ù $ (.5) and for our components B œ B C œ B D œ B " # $ (.6) In this notation, e ould express the vector lvù ith the notation here the lvù œ B l/ Ù B l/ Ù B l/ Ù " " # # $ $ (.7) -component of the vector is given by B " B œ Ø/ lvù " " (.) The notation Ø/ " lvù must be equivalent to a dot product. The unit orthogonal vectors all satisfy an equation of the form Ø/ / Ù œ $ (.9) For an R-dimensional space e simply allo the subscripts to range from 1 to R. Thus, the -dimensional vector lαù can be expressed by the equation l Ù œ + l/ Ù + l/ Ù â + l/ Ù œ + l/ Ù (.10) α " " # # 3 3 3œ" here the + 3 = are simply scalar coefficients (hich could be complex), and the l/ 3Ù's are orthoganal unit vectors. Once a set of unit vectors has been established, the vector can be represented solely based upon its components, i.e., lαù œ + " ß+ # ßáß+ (.11) just as you might designate the location of a particle based solely upon its coordinates BßCßD in a predefined coordinate system.

3 Quantum Mechanics (Griffith) 3 This looks very similar to the the situation e have hen e solve the infinite square ell problem. There are an infinite (though countable) set of eigenfunctions from hich e construct the most general ave function solution. We might rite this out symbolically as < Ù œ + < Ù (.12) The individual coefficients can be determined by utilizing the analogy of the dot product of the unit vector ith the vector <Ù Ø< < Ù œ + Ø< < Ù œ + $ œ + (.13) We represented the orthonormality condition for the eigenfunctions of the infinite square ell by the equation 7 7 ( < ÐBÑ< ÐBÑ.B œ $ (.14) It should be obvious that the dot product notation is similar to the integral orthonomality conditions, or that 7 Ø< 7 < Ù œ ( < ÐBÑ< ÐBÑ.B œ $ 7 (.15) This is a valuable short-hand ay of riting out some of our integral expressions. Hoever, there is a slight difference beteen these to notations hich e ill discuss later on. Linear Vector Spaces Linear vector spaces are defined ith regard to the ay these vectors can be manipulated mathematically. We define the vector addition of any to vectors lαù and l" Ù by the equations and lα Ù l" Ù œ # Ù (.16) lα Ù l" Ù œ + ", " ß+ #, # ßáß+, (.17) This means that the result of adding any to vectors is itself a vector. Vector addition is both commutative (i.e., you can reverse the order of addition ithout changing the result) lα Ù l" Ù œ l" Ù lα Ù (.1) and associative lα Ù l" Ù l# Ù œ lα Ù l" Ù l# Ù (.19)

4 Quantum Mechanics (Griffith) 4 We define the null vectorl!ù as that vector satisfying the equation lαù l!ù œ lα Ù (.20) so that l!ù œ!ß!ßáß! (.21) and the inverse or negative or a vector lαù by the vector l αù hich satisfies the equation lαù l α Ù œ l!ù (.22) so that l αù œ + " ß + # ßáß + (.23) Scalar multiplications is defined as the multiplication of a scalar times a vector. This product is just another vector defined by the equation or -lα Ù œ l# Ù (.24) -lαù œ -+ " ß-+ # ßáß-+ (.25) Obviously the multiplication of the scalar " times a vector gives the inverse of that vector. Scalar multiplication is distributive ith respect to vector addition and ith respect to scalar addition + lα Ù l" Ù œ +lα Ù +l" Ù (.26) It is also associative ith respect to multiplication of scalars +, lαù œ +lαù,lα Ù (.27) +,lαù œ +, lα Ù (.2) In three dimensions e define to types of vector multiplication - the dot product and the cross product. Only the dot product (called the inner product), hoever, has a natural extension into multi-dimensions. In three dimensions, the inner product of to vectors E F t t œ lellfl t t cos ) (.29) is a scaler hich can be thought of geometrically as the length of one vector multiplied by the projection of the other vector onto the first, as seen in the diagram belo. If the to vectors Et and Ft are perpendicular to each other, then E F t t œ!, and if they are parallel to each other, and of unit length, then E F t t œ lellfl t t. This means of course that the

5 Quantum Mechanics (Griffith) 5 absolute magnitude of a vector can be expressed as lel t œ È E E t t (.30) A From this geometrical definition, e can see that θ E F t t lellfl t t B œ cos ) Ÿ " (.31) or, in our ne notation, Ø+,Ù ÈØ++ÙØ,,Ù Ÿ " (.32) Although the inner product Ø+,Ù may be complex in general, the absolute value of this last equation gives hich is knon as the Schartz inequality. # lø+,ùl Ÿ Ø++ÙØ,,Ù (.33) The Formal Structure Of Quantum Mechanics In previous chapters, e have looked at the solutions to several one-dimensional problems. In each case, e started ith the time-dependent Schrodinger equation # # h < ÐBß>Ñ ZÐBÑ< ÐBß>Ñ œ 3h < ÐBß>Ñ #7 B# >.34 hich can be ritten in operator form as here the Hamiltonian operator Ls is given by Ls < ( Bß>Ñ œ Is < ÐBß>Ñ.35 Ls œ # :s #7 ZÐBÑ s and the total energy operator Is is given by Is œ 3hÎ>Þ.36

6 Quantum Mechanics (Griffith) 6 If the potential energy is time-independent, e shoed that the solution to the time-dependent Schrodinger equation is given by here < ÐBß>Ñ œ < ÐBÑ/ 3I>Îh.37 <ÐBÑ is the solution to the time-independent Schrodinger equation # # h #7 B This last equation can also be expressed in operator form as # < ÐBÑ ZÐBÑ< ÐBÑ œ I< ÐBÑ.3 Ls < EÐBÑ œ I< EÐBÑ.39 hich is the so-called energy eigenvalue equation. The subscript on the eigenfunction is there to remind us that the eigenfunctions are functions of the energy of the system, and that for each different value of the energy, e have a different eigenfunction. In some cases the energy spectrum is discrete (such as the infinite square ell and the simple harmonic oscillator), but in others the spectrum is continuous (such as the free particle). The most general solution to the Schrodinger equation (since it is a linear equation) is the sum (or integral) over all possible states, or G ÐBß>Ñ œ E < ÐBß>Ñ œ 3 E < ÐBÑ/ = >.40 here e have explicitly ritten the expansion as if it ere of a discrete set of functions ith œ I Îh. In the case here the energies form a continuum, e rite = + G ÐBß >Ñ œ ( 1ÐIÑ< IÐBÑ/.I.41 3 E > / h In both of these expansions, it is assumed that the eigenfunctions (both discrete and continuous) form a complete set of orthogonal solutions so that any vector in the vector space can be represented by a linear combination of these basis vectors. For the case here the energies are discrete, the orthonormality condition is expressed by the equation ( < ÐBÑ< ( B ).B œ $.42 and for the case of continuous energy levels, by 7 7ß ( < ÐBÑ< I( B ).B œ $ ÐI I Ñ.43 I A short-hand notation (called the bracket, or bra ket notation) as introduced by P. A. M. Dirac to help simplify these complex expressions. (Hoever, there is the possibility of sometimes getting lost in the simplification!) To understand this notation,

7 Quantum Mechanics (Griffith) 7 e begin ith an arbitrary eigenvalue equation hich may be ritten as U? s ÐBÑ œ ;? ÐBÑ.44 here? ÐBÑ is an eigenfunction of the operator U s ith corresponding eigenvalue ;. Dirac proposed that e rite an eigenvector in terms of the eigenvalues associated ith that eigenvector, i.e., Ul; s Ù œ ; ; Ù.45 Therefore, e can rite the eigenvalue equation of the Hamiltonian Ls liù œ IlIÙ.46 here liù is an eigenvector of enegy. (The implication of the subscript is that this eigenvalue equation forms a discrete set of eigenvectors, and this ill be our assumption throughout our development of the Dirac notation.) We rite the expression for representing the orthonormalization condition by ØI7lIÙ œ $ 7.47 here ØI is understood to be the complex conjugate of IÙ, and here the combination of the to is similar to our normalization integral. The complex conjugate vector ØI is called the bra, hile the eigenvector IÙ is called the ket. The most general solution to the time-dependent Schrodinger equation, in terms of the eigenfunctions of position, is given by G ÐBß>Ñ œ 3 E < ÐBÑ/ = >.4 If e associate the time-dependence ith the expansion constant, as follos e can rite our expansion of the eigenfunctions in the form EÐ>Ñ œ E / 3 = >.49 G ÐBß>Ñ œ EÐ>Ñ< ÐBÑ.50 The Dirac notation for eigenvectors is similar in form to this equation, and is given by lgð>ñù œ E > liù.51 here the time-dependent notation in the ket tips us off that the expansion constants contains the time factor! As mentioned earlier, Dirac's notation for the combination of eigenvectors is not quite equivalent to the position-time equations hich e have been using. You may remember that e stated in an earlier chapter that the state of the system could be represented either in position (configuration) representation or in momentum

8 Quantum Mechanics (Griffith) representation. This is specifically denoted in the Dirac notation in the folloing ay. The state of the system expressed in configuration representation and in momentum representation are denoted, repectively, by G ÐBß>Ñ ØBl< Ð>ÑÙ.52 and 9 Ð:ß>Ñ Ø:l< Ð>ÑÙ.53 in Dirac notation. The ket < (t) Ù is not equivalent to G( Bß> ), but is even more fundamental. This ket is called the state vector of the system. This state vector may be projected onto position space by taking a dot product of < > Ù ith BÙ, an eigenvector of the position operator Bs! It may likeise be projected onto momentum space by taking a dot product of l< Ð>ÑÙ ith l:ù, an eigenvector of the momentum operator s:þ (The state vector may be projected onto any other basis set as ell.) In a similar manner e ill take the position representation of the eigenfunctions of the Hamiltonian to be < ÐBÑ ØBlI Ù.54 and e ill represent the complex conjugate of this eigenfunction ith the notation < ÐBÑ ØIlBÙ.55 We have previously expressed the normalization condition for the energy eigenstates in position representation by the equation ( < ÐBÑ< ÐBÑ.B œ $.56 In Dirac notation, this equation becomes 7 7 ( ØI7lBÙØBlIÙ.B œ $ 7.57 But e have already stated that the orthonormality condition for the eigenvectors satisfies the equation This means that the integral over B given by must someho act like a unity operator. ØI7lIÙ œ $ 7 œ ( lbùøbl.b œ ".5

9 Quantum Mechanics (Griffith) 9 Examples of Dirac Notation To gain some familiarity ith the use of the Dirac notation, let's examine the expectation value of the energy, ØLÙ s. The eigenvalue equation for the Hamiltonian is given by Ls liù œ IlIÙ.59 here the energy eigenvectors are not representation-dependent. No, the expectation value of the Hamiltonian for a system in a given eigenstate of that system is given by ØLÙ s œ ØI lls li Ù œ I ØI li Ù œ I $.60 regardless of the specific representation hich is used! Since the energy eigenvectors derived from the time-independent Schrodinger equation form a complete set of vectors, e can express a general state of the system in terms of these eigenvectors according to the equation The expectation value for the energy in this case is given by l< Ð>ÑÙ œ EÐ>ÑlIÙ.61 ØLÙ s œ Ø< (t) lls l< (t) Ù= E Ð>Ñ ØI Ls 7 7 EÐ>ÑlIÙ.62 7 No, since the coefficients are just constants times the time-factor, and since Ls is not explicitly time-dependent, the Hamiltonian operator simply acts on the eigenvector liù, giving ØLÙ s œ E Ð>ÑE Ð>ÑØI lls 7 7 liù 7ß œ E7Ð>ÑEÐ>ÑIØI7lIÙ 7ß œ E7Ð>ÑEÐ>ÑI$ 7 7ß œ # le Ð>Ñl I So e see that the expectation value of the Hamiltonian is just the eighted average of the energies of the different states of the system under study. In the same ay, e can sho that the coefficients are probability amplitudes for finding the system in a particular state of the system. We can see this by looking at the

10 Quantum Mechanics (Griffith) 10 normalization condition as it is applied to a general state of the system. We have Ø< (t) l< (t) Ù œ E7Ð>ÑØI7 EÐ>ÑlIÙ.64 7 œ E Ð>ÑE Ð>ÑØI li Ù 7ß 7 7 œ E Ð>ÑE Ð>Ñ $ 7ß 7 7 œ # le Ð>Ñl œ " 7 7 This shos that the sum of the probability of finding the system in each particular eigenstate of the system is alays equal to unity. This ill be true even if the system is in a single eigenstate, say li; Ù, in hich case the coefficient E; ould be unity. We next consider ho e are to find the individual coefficients for the expansion of a general avefunction. The general expansion is given, as e stated above, by the equation l< Ð>ÑÙ œ EÐ>ÑlIÙ.65 If e no take the dot-product of our general vector onto a particular energy eigenvector (hich represents one of the basis vectors hich span the solution space of our particular problem) e obtain ØI l< Ð>ÑÙ œ E Ð>ÑØI li Ù 7 7 œ E Ð>Ñ$ œ E Ð>Ñ This means e can express the general ave equation as l< Ð>ÑÙ œ EÐ>ÑlIÙ œ ØI l< Ð>ÑÙlI Ù.67 This last equation can be ritten in a slightly different fashion as: l< Ð>ÑÙ œ liùøil l< Ð>ÑÙ.6

11 Quantum Mechanics (Griffith) 11 Written in this ay, e see that the expression in parentheses is a special kind of unity operator œ liùøil œ ".69 much like the unity integral e introduced above. This last equation is called the closure relationship. It implies that the entire solution space is spanned by the set of basis vectors li Ù. Next, e examine the Dirac notation for the free-particle ave function in terms of the various energy (or momentum) components. We have shon earlier that the most general form for the free-particle avefunction must be given by 3:BÎh / < ÐBß>Ñ œ ( 9Ð:ß>Ñ.: È# 1h.70 so that the avefunction is normalizable. We can express the position dependent and the momentum dependent avefunctions, respectively, in Dirac notation using the equations < ÐBß>Ñ œ ØBl< Ð>ÑÙ 9 Ð:ß>Ñ œ Ø:l< Ð>ÑÙ.71 No, if e begin ith a simple statement of equality of the avevector ith itself, e have l< Ð>ÑÙ œ l< Ð>ÑÙ ØBl< Ð>ÑÙ œ ØBl< Ð>ÑÙ ØBl< Ð>ÑÙ œ ØBl l< Ð>ÑÙ ØBl< Ð>ÑÙ œ ( ØBl:ÙØ:l< Ð>ÑÙ.: here e have used the unity operator in momentum space.72 ( l:ùø:l.:.73 This definition is analogous to the definition e stated earlier for position space and for the energy eigenvectors, liùøil.74 You ill notice that for eigenvectors hich take on discrete values e utilize a summation, hile for eigenvectors hich take on a continuous range of values e utilize

12 Quantum Mechanics (Griffith) 12 an integral. Thus, comparing ØBl< Ð>ÑÙ œ ( ØBl:ÙØ:l< Ð>ÑÙ.:.75 ith our previous expression for the free particle ave function 3:BÎh / < ÐBß>Ñ œ ( 9Ð:ß>Ñ.: È# 1h.76 hich e no rite as 3:BÎh / ØBl< Ð>ÑÙ œ ( Ø:l< Ð>ÑÙ.: È# 1h.77 e see that e must identify the expression ØBl:Ù, hich is the projection of the momentum eigenvector onto the position eigenvector, ith the equation 3:BÎh ØBl:Ù œ / È# 1h.7 This also implies that 3:BÎh ØBl:Ù œ Ø:lBÙ œ / È# 1h.79 In orking ith Dirac notation, the expressions ØBBÙ and Ø:: Ù ill often occur. We no ant to determine just hat these expressions are equivalent to. We begin ith the expression ØBBÙ and insert the momentum unity operator, obtaining Thus, ØBBÙ = ( ØB:ÙØ:BÙ.: (.0) = ( = ( e e È21h È21h.: e 21h = $ ( B B ) i :B/ h -i:b / h i : ( B - B )/ h.: ØBBÙ $ ( B B ) (.1)

13 Quantum Mechanics (Griffith) 13 By a similar process e can sho that Likeise, e could sho that Ø:: Ù $ (: : ) (.2) No, let's examine the expectation value for the momentum using Dirac notation. We begin ith the expectation value defined by the equation Ø:Ù = Ø < ( > ) s: < ( > ) Ù (.3) Utilizing to distinct unity relationships for the momentum, the expectation value can be expressed by Ø:Ù = Ø < ( > ) s: < ( > ) Ù = ( Ø < ( > ) :ÙØ::: s ÙØ: < ( > ) Ù.:.: (.4) No e kno the result of s: operating on the eigenvector :Ù: it's just the value of the momentum, :, or Thus, the expectation value becomes s : : Ù = : : Ù (.5) Ø:Ù = Ø < ( > ) s: < ( > ) Ù = ( Ø < ( > ) :Ù : Ø:: ÙØ: < ( > ) Ù.:.: (.6) Using the identity Ø:: Ù = $ (: : ), this becomes or hich is equivalent to Ø:Ù = Ø < ( > ) s: < ( > ) Ù = ( Ø < ( > ) :Ù : $ (: : ) Ø: < ( > ) Ù.:.: (.7) Ø:Ù = Ø < ( > ) s: < ( > ) Ù = ( Ø < ( > ) :Ù : Ø: < ( > ) Ù.: (.) Ø:Ù = ( 9 (:, > ): 9( :, > ).: (.9) This is the expectation value of the momentum as represented in momentum representation (or momentum space).

14 Quantum Mechanics (Griffith) 14 To obtain the corresponding expression in configuration space, e make use of to distinct unity operators in position space Ø:Ù = Ø < ( > ) s: < ( > ) Ù = ( ( Ø < ( > ) BÙØB:BÙØB s < ( > ) Ù.B.B (.90) In this equation, e have the quantity ØB:BÙ s. But e don't kno the result of s: operating on the eigenvector lbù. Hoever, e can make use of the unity operator in momentum space and rite ØB:BÙ s œ ( ØB:: s ÙØ: BÙ.: (.91) œ ( : ØB: ÙØ: BÙ.: No, the projection of the momentum vector onto position is given by 3:BÎh ØBl:Ù œ / (.92) È# 1 h and the derivative of this function ith respect to position gives 3:B/ h 3:B/ h e 3: e 3: B œ œ ØBl:Ù È 21h h È21h h (.93) Similarly, 3:B/ h 3:B/ h e 3: e 3: œ œ Ø:BÙ B È 21h h È21h h so e can rite : ØBl: Ù œ i h ØBl: Ù (.94) B : Ø: lbù œ i h Ø: lbù B (hich means than the number : is real!). ØB:BÙ s = ( ØB:Ù Œi h Ø:BÙ.: (.95) B

15 Quantum Mechanics (Griffith) 15 Substituting this back into the equation for the expectation value of : and separating out the dependence on B e have Ø:Ù = Ø < ( > ) s: < ( > ) Ù = ( ( Ø < ( > ) BÙØB:Ù œ( Œi h Ø:BÙ ØB < ( > ) Ù.B.:.B B We must integrate the term in braces by parts ith respect to B ( i h Œ Ø:BÙ ØB < ( > ) Ù.B B The avefunction œ 3h Ø:BÙØB < ( > ) Ùº ( Ø:BÙ Œ ØB < ( > ) Ù.B B Ÿ ØB < ( > ) Ù goes to zero at infinity, so that this equation becomes (.96) ( i h Œ Ø:BÙ ØB < ( > ) Ù.B œ ( Œ 3h ØB < ( > ) Ù Ø:BÙ.B (.97) B B The expectation value of : can no be expressed by the equation Ø:Ù = ( ( ( Ø < ( > ) BÙØB:Ù Š i h ØB < ( > ) Ù Ø:BÙ.B.:.B (.9) B With a slight rearrangement, e rite Ø:Ù = ( ( ( Ø < ( > ) BÙØB:ÙØ:BÙ Š i h ØB < ( > ) Ù.B.:.B (.99) B We have eliminated all dependence upon the momemtum, except for hat you ill recognize as the unity operator in momentum space. This equation, then reduces to Ø:Ù = ( ( Ø < ( > ) BÙØBBÙ Š i h ØB < ( > ) Ù.B.B (.100) B Here e recognize ØBBÙ = $ ( B B ), so that integration over B finally gives Ø:Ù = ( < ( B>, ) Š i h < ( B>, ).B (.101) B our previous result for the expectation value of the momentum in configuration space. Hermetian Operators in State Vector Notation The basic interpretation of the ave function as it relates to the probability of measuring a particle's position, momentum, etc., requires us to normalize the ave

16 Quantum Mechanics (Griffith) 16 function. This same condition must also apply to the avevector for a state of the system giving Ø < ( > ) < ( > ) Ù œ 1 (.102) This normalization condition must also be time-independent (provided the particles are not created or destroyed), so that > In our ne notation this last equation becomes Ø < ( > ) < ( > ) Ù Ÿ = 0 (.103) Œ Ø < ( > ) l < ( > ) Ù Ø < > Œ l < > Ù (.104) > > The Schrödinger equation in our ne notation can be expressed as L s < > Ù œ I s < > Ù L s< > Ù œ 3h < > Ù > ØL s< > œ 3h Ø < > > (.105) here Š L s< > Ù œ ØL s< > Þ Note: When an operator b s operates on a state vector < Ù it produces a ne state vector hich is represented by the equation This means that e can rite Our requirement of normalization, therefore, becomes or bs < Ù œ bs < Ù (.106) " < > Ù œ L s< > Ù (.107) > 3h " Ø < > œ ØL s< > > 3h " ØL s " < > < > Ù Ø < > L s< > Ù œ! (.10) 3h 3h Ø < > L s< > Ù œ ØL s< > < > Ù (.109)

17 Quantum Mechanics (Griffith) 17 Since the complex conjugate is represented by reversing the direction of the state vectors e can rite this last equation as Ø < > L s< > Ù œ ØL s< > < > Ù œ Š Ø < > L s< > Ù (.110) hich is often expressed in the more compact form ØLÙ s œ ØLÙ s (.111) This means that the expectation value of the Hamiltonian is real! Just as e pointed out earlier, for any physically measurable quantity the expectation value must be real. An operator hich satisfies this condition is called an Hermitian operator. Thus, all physically measurable quantities must be represented by Hermitian operators hose expectation values are real! But this does not mean that all useful quantum mechanical operators are Hermitian. Hermitian Operators and the Adjoint For any operator Us, e define the adjoint of that operator Us according to the folloing equation Ø < ( > ) Ul s < (t) Ù œ Ø < ( > ) Us < (t) Ù œ ØUs < (t) < (t) Ù œ Ø < ( > ) Us < (t) Ù (.112) Let's compare this definition ith hat e learned about the Hamiltonian operator. First, using the definition of the adjoint, e obtain Ø < ( > ) Ll s < (t) Ù œ Ø < ( > ) Ls < (t) Ù = ØLs < (t) < (t) Ù œ Ø < ( > ) H s < (t) Ù (.113) But from hat e learned above, Ø < > L s< > Ù œ ØL s< > < > Ù œ Š Ø < > L s< > Ù (.114) This obviously means that the adjoint of the Hamiltonian operator must be equal to the Hamiltonian operator itself. The Hamiltonian is said to be self-adjoint. This must be true for all Hermitian operators hich represent physically measureable quantities. This means that all Hermitian operators are self-adjoint! Problem.1 Sho that the general definition of the adjoint of an operator Es is given by the equation " # " # # " Ø9 les 9 Ù œ ØEs 9 l9 Ù œ Ø9 les 9 Ù Start ith the definition of the adjoint operator operating on a vector <Ù composed of to other vectors such that < Ù œ + " 9" Ù + # 9# Ù. Problem.2 An operator Ss can be formed from a combination of Hermitian operators. This operator

18 Quantum Mechanics (Griffith) 1 may or may not be Hermitian. Determine the necessary conditions for the folloing operators to be Hermitian: a) The operator Gs is a linear combination of Hermitian operators Es and Fs Gs œ α E s " Fs Determine the necessary conditions on the constants α and ". b) The operator Hs is a product of to Hermitian operators Es and Fs Hs œ EF s s Problem.3 The so-called step-up and step-don operators " s+ œ s: 37 Bs È#7 e = f " s+ œ s: 37 Bs È#7 e = f are utilized in solving the quantum oscillator problem using operator methods. (a) Determine the adjoint of both of these operators. (b) Are these to operators Hermitian? Commutator Relationships and Measurement Theory When to operators operate on a ave function, the order of operation may be important. As e have demonstrated earlier, if e operate on an arbitrary function 0 ÐBÑ ith B s and then s: expressed in position representation, e obtain 0 ÐBÑ ss :B0ÐBÑ œ 3h B0ÐBÑ œ 3h0ÐBÑ 3hB (.115) B B hereas if e operate on this same function first ith the s: operator and then the Bs operator, e have 0ÐBÑ B:0ÐBÑ ss œ BŒ 3h (.116) B Clearly, these to operations are not identical. We define the commutator cbß: s sd of to operators B s and s: by the equation cbß: s sd B: :B ss ss (.117) If this commutator ere zero, then the to operators ould commute, i.e., it ould make no difference in hat order they operate - you ould get the same results either ay. Clearly, the operators sb and s: do not commute. The value of the commutator is determined by alloing the commutator to operate on an arbitrary function. From our previous results, e can see that

19 Quantum Mechanics (Griffith) 19 cbß: s sd0ðbñ œ ss B:0ÐBÑ :B0ÐBÑ ss 0ÐBÑ 0ÐBÑ œ BŒ 3h Œ 3h0ÐBÑ 3hB B B œ 3h0ÐBÑ (.11) so that e have for the commutator of B s and s: : cbß: s s d œ 3h (.119) Notice that the commutator cs:ßb s d œ 3h (.120) The commutation properties of operators are extremely important. As e mentioned previously, these commutation relationships are valid no matter hat representations you are using, position or momentum, and therefore the commutation relationships must be very fundamental in nature. The commutator relationships are valid for state vectors as ell. Problem.4 Sho that the commutator csbß: sd gives the same result hen operating on the function 0 B in position representation, or on the function g : in momentum representation. Notice that the non-commutativity property of the operators sb and s: is of the order of h, i.e., it is extremely small. This means that the order of operation (or measurement) is unimportant hen h is so small that it is neglegible - i.e., in the classical limit. But for quantum systems the order of operation is important. The fact that sb and s: do not commute is linked to the uncertainty principle. In general, one can sho that # # Š? Es Š? Fs " º Ø Es ßFs Ù % º # (.121) here # # Š?E s œ ØŠ E s ØEÙ s # Ù œ ØE s Ù ØEÙ s # (.122) is the root-mean-square deviation in the measurable quantity designated by E. Thus, if to operators commute, there is no uncertainty in the respective measurements. For position and momentum e obtain # # " " h? sb? s: º Ø sb ß: s Ù œ l3hl œ % c d # º % % # # (.123) hich gives us? sb? s: hî# (.124)

20 Quantum Mechanics (Griffith) 20 As you ork ith commutators you may find the folloing relationships useful c0 sb ß: sd œ 3h 0ÐBÑ s (.125) B s csbß1ð:ñ s d œ 3h 1Ð:Ñ s : s csbß0 sb d œ cs:ß1 s: d œ! Problem.5 Prove the commutation relations given above. You may do these in position or momemtum representation. Time-Dependent Expectation Values and the Correspondence Principle In ave vector notation, the expectation value of an operator E s is given by Ø < > E s < > Ù (.126) If e take the time derivative of this expectation value e obtain..> Ø < > E s < > Ù œ Œ > Ø < ( > ) E s < > Ù â (.127) Ø < > E s < > Ù Ø < > E s Œ < > Ù > > Just as e did hen e examined the normalization condition, e use Schrödinger's equation to rite out the time derivative of the ave vectors L s < > Ù œ I s < > Ù (.12) L s< > Ù œ 3h < > Ù > ØL s< > œ 3h Ø < > > Thus, the time derivative of the expectation value of E s can be ritten as..> Ø < > E s < > Ù œ Œ 3 h ØL s< > E s < > Ù â (.129) 3 Ø < > E s < > Ù Ø < > Es Œ L s< > Ù > h

21 Quantum Mechanics (Griffith) 21 Factoring out the constant and utilitzing the fact that the Hamiltonian is self-adjoint, e can simplify this expression to obtain. Ø < > E s < > Ù œ Ø < > E s < > Ù 3 Ø < > Š LE ss EL ss < > Ù (.130).> > h hich can be ritten utilizing the commutation relationship. Ø < > E s < > Ù œ Ø < > E s < > Ù 3 Ø < > Ls ßE s < > Ù (.131).> > h This last equation is an extremely useful relationship. It states that if the operator E s is not explicitly a function of time, and if it commutes ith the Hamiltonian operator, then the expectation value of the operator is a constant - it does not change in time. Examples Let's look at the time-derivative of the expectation value of position. Using the equation above, e obtain. Ø < > sb < > Ù œ Ø < > sb < > Ù 3 Ø < > Ls ßsB < > Ù.> > h This is often ritten ithout the explicit representation of the ave vector, as We pointed out earlier that the commutator..> Ø s BÙ œ Ø s BÙ 3 Ø > h L s ß s B Ù # # L s s s: B œ s s s: ß ZÐBÑ ßB œ ßB s czðbñßb #7 #7 s s d The second term on the right is zero, and the first term can be ritten as # # # s : ßB œ B s: s sß œ 3h s : œ 3h s: #7 #7 s: #7 7 So e have, for the time derivative of the expectation value of the position operator. 3 s: Øs: Ù ØsB Ù œ! Ø 3h Ù œ.> h 7 7 Problem.6

22 Quantum Mechanics (Griffith) 22 Using this same proceedure, sho that. BÑ.> Ø:Ù œ Ø ZÐs s B s Ù hich is compatible ith Neton's second la. Problem.7 Sho that the time derivative of the expectation value of the Hamiltonian is zero, unless the Hamiltonian is explicitly time-dependent. This is just a statement of the conservation of energy. Measurement and Complete Sets of Basis Vectors We interpret the equation E+Ù s œ ++Ù as representing the process of measuring a physical quantity represented by E s (for example the position, or momentum, or spin, or angular momentum, or color, or temperature, or hatever). This equation indicates that hen e make such a measurement on an eigenvector of the operator e obtain a certain value corresponding to that particular eigenstate and no other. That is, hen e make a measurement of position, or momentum, or spin, or angular momentum, or color, or temperature - e obtain a certain value depending upon the particular state of the system. Similarly, if e represent the process of measuring a separate physical quantity by the operator F s, e rite F,Ù s œ,,ù No, let's assume that e have a quantum system in some state <Ù hich is simultaneously an eigenvector of the to operators Es and Fs. We can, therefore, express this eigenvector more specifically by riting <Ù œ +,Ù here e kno that E+,Ù s œ ++,Ù F+,Ù s œ,+,ù If e no make a measurement of the state +,Ù corresponding to the operator E s and then follo that ith a measurement corresponding to the operator F s, e have F s se +,Ù œ F s + +,Ù œ +F+,Ù s œ +,+,Ù

23 Quantum Mechanics (Griffith) 23 If e make our measurements in the reverse order, e obtain EF+,Ù s s œ E s, +,Ù œ,e+,ù s œ,++,ù No, since + and, are just numbers associated ith a measurement, and since the numbers commute, the order of the operations are unimportant - i.e., the operators E s and F s commute, or Es ß Fs œ! Thus, if to operators commute, e can describe the state of the system in terms of the to eigenvalues associated ith these operators. For example, in the case of the free particle the Hamiltonian commutes ith the momentum operator. This means that a state vector designated only by its energy eigenvalue < I Ù is degenerate, i.e., there are actually to different momentum states hich have the same energy; one for each direction of the momentum of the particle. We can remove this degeneracy by designating the state vector < IßT Ù. Thus, to completely specify the state of a system one must find all operators hich commute. A degeneracy in any observable quantity implies that there is some other measurement hich can be made hich ill unambiguoously define the quantum state of the system.