QUANTUM MECHANICS, BRAS AND KETS

Transcription

1 PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented b a state vector or ket and t s denoted b a smbol such as. Lke an ordnar vector, the ket can be epressed as a lnear combnaton of bass kets wth coeffcents c : The bass ket c. s often abbrevated smpl b. The correspondng bra s c where c s the comple conjugate of c. The ket s an abstract quantt that represents a quantum state. Often, we wsh to be more specfc an eplct representaton that gves us nformaton about (for eample) spatal (poston) dstrbuton of electron denst. To do ths, we project the ket onto the bras that are the egenstates of poston. The resultng functon s called the wave functon (n the poston representaton). An eplct eample of a set of bass wave functons are the hdrogen ground state orbtals used to develop the states of the datomc molecule, lnear chan, etc.: / ( r / a ( r ) e 3 a 4) where a s the Bohr radus and r s the radal dstance of the electron from the th nucleus. (We wll rarel actuall use the eplct forms of these bass functons. Rather we make use of ther mathematcal propertes such as orthogonalt and normalaton see below.) Projectons scalar products: The coeffcents n the lnear combnaton are the projectons of on the bass states,.e. c where the object s a scalar product analogous to the dot product of ordnar vectors. The scalar product of two state vectors (or the projecton of one on the other) s a sum of products of ther respectve projectons on the bass (agan just lke ordnar vectors: a b a b + a b + a b ). Thus,. If we were usng eplct functons of spatal coordnates to represent the states and, we would evaluate the projecton wth an ntegral ( r ) ( r ) dv. all space

2 PH575 SPRING The quantt s often called the overlap ntegral. It represents how much of one functon s contaned n another. The overlap ntegral of orthogonal functons (see below) s ether ero or one. The orthonormal propert usuall assumed for the bass states can be epressed n terms of ther scalar products: j f j; j f j. Ths same statement s often wrtten n terms of the Kronecker delta: j j State vectors are usuall normaled:. Ths leads to the result that The quantt c c c c s the probablt that n a measurement, an electron wll be found n the partcular bass state. The normalaton condton smpl epresses the fact that the electron must be found n some bass state, thus the sum of all probabltes equals. Operators In quantum mechancs, observable phscal quanttes are represented b operators. There are two cases to consder: Case I: An operator L operates on a state vector and elds the same state vector smpl multpled b a constant: L CL. In ths case, s an egenvector of the operator L and the constant C L s an egenvalue. The most mportant eample of ths s gven b the Schrödnger equaton H E where the Hamltonan (energ operator) s, n Cartesan spatal coordnates, ( ) ) ) H knetc + potental energ + V(,, ) + V(,, ) m + + ) ) ) m A state vector that satsfes the Schrödnger equaton s an energ egenfuncton or egenvector wth egenvalue equal to the energ of the state E. An mportant part of the determnaton of the electronc structure of solds s to determne the energ egenvalues for partcular assembles of atoms. Case II: The state vector s NOT an egenvector of the operator L. In ths case the operaton of L on produces a dfferent state vector or lnear combnaton of state vectors, denoted here b : L constant.

3 PH575 SPRING The matr elements of an operator L are epressed n terms of a partcular bass set. The are scalar products of a bass bra j and the ket produced b the acton of L on,.e. j L L. j When j, the dagonal matr element L L s the epectaton value of the phscal quantt represented b L n the partcular bass state. If, n addton, s an egenvector of L, then L s just the egenvalue C L. For eample, f L s the energ operator H and our bass states are the atomc ground states, then H s the energ of an electron n the ground state of atom. Specfc eamples of the above wth reference to the hdrogen atom atomc orbtals. State of a phscal sstem: Suppose the state of an electron n a H-atom s an unequal superposton of the s and p states. We denote ths state vector or ket b. can be epressed as a lnear combnaton of bass kets, whch s ths case are the egenstates of the H-atom Hamltonan. In ths case the bass kets are s and p. The coeffcents c n ths case are not equal, and for llustraton are set n a : rato. Proper normalaton requres that the squares of the coeffcents sum to : c 5 s + 5 p. The bass kets could also be labeled b the quantum numbers, n, l, m l, of the state: and. The correspondng bra s c where c s the comple conjugate of c. If we want an eplct representaton that gves us nformaton about spatal (poston) dstrbuton of electron denst, we project the ket onto the bras that are the egenstates of poston. The resultng functons s called the wave functons (n the poston representaton). Here, the bass wave functons are the hdrogen orbtals: s (r,,) 3 a er/a ( 4 / ( 4, p (r,,) a 3 ( ) e r/a where a s the Bohr radus and r s the radal dstance of the electron from the nucleus. / Projectons scalar products: Project the state onto the s bass state: The notaton s ths: s Project the state onto the p bass state: The notaton s ths: p

4 PH575 SPRING Projectons are reall ntegrals. The notaton ths case n 3 spatal dmensons: the functon r,, conjugate functon s s means ntegrate over all space, n...d dd. The notaton means nsert v ( ) nto the ntegral and the notaton s means nsert the comple ( r,, ) nto the ntegral. Thus: s ( r,, ) ( r,, )r sn dr d r d. Because we can look up the poston representaton of bass wave functons, we can plug n and evaluate. But because the bass functons of the Hamltonan are orthogonal to one another, the ntegraton s not necessar (but ou should check that t actuall does evaluate to or ero) s s 5 s + 5 p s s s p 5 p s 5 s + 5 p 5 p s + 5 p p 5 The scalar product of two state vectors (or the projecton of one on the other) s a sum of products of ther respectve projectons on the bass (agan just lke ordnar vectors: a b a b + a b + a b ). Suppose another state of the H-atom s represented b s + 3p. Then the projecton of onto s 5 s + 5 p s + 3p s s + s 3p + p s + p 3p. Or we could evaluate the projecton wth the ntegral ( r ) ( r ) dv. all space The quantt s often called the overlap ntegral. It represents how much of one functon s contaned n another. The overlap ntegral of orthogonal functons (see below) s ether ero or one. Operators Case I: The Hamltonan operator H m e operates on a state vector 4 r and elds the same state vector smpl multpled b a constant. Eamples are the

5 PH575 SPRING states s, p 3p H s e a s ; H p e a ( ) p ; H 3p e a 3 ( ) 3p In ths case s, p 3p are egenstates of the operator H, whch s wh we chose them as the bass states. The egenvalues are the energes of the respectve orbtals. Case II: The Hamltonan operator H m e 4 r operates on a state vector and elds a dfferent state vector (multpled b some constant). Eamples are and, whch are NOT egenvalues of H. For eample, H e 5a where the ket s found below: H H 5 s + 5 p e 5a s + e 5a ( ) p e s + 5a 4 p The matr elements of the Hamltonan H are epressed n terms of a partcular bass set. The are scalar products of a bass bra j and the ket produced b the acton of H on,.e. j H H j. If the bass vectors are egenfunctons, the matr s dagonal. For eample: H s,s s H s E s s s E s H s,s s H s E s s s E s. The matr s (n part t s actuall nfnte n dmenson) H s s p p p s s p p p E E E E E E 3 The dagonal matr element L L s called the epectaton value of the phscal quantt represented b L n the partcular bass state.

6 PH575 SPRING Usng the same bass states to epress the L operator, we fnd the matr s NOT dagonal because the p and p states are not egenfunctons of the L operator: L s s p p p s s p p p