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Lecture 13: Angular Momentum-I. The material in this lecture covers the following in Atkins. Rotational Motion Section 12.6 Rotation in two dimensions Lecture on-line Angular Momentum in 2D (PowerPoint) Angular Momentum in 2D (PDF format) Handout for this lecture Tutorials on-line Vector concepts Basic Vectors More Vectors (PowerPoint) More Vectors (PDF) Basic concepts Observables are Operators - Postulates of Quantum Mechanics Expectation Values - More Postulates Forming Operators Hermitian Operators Dirac Notation Use of Matricies Basic math background Differential Equations Operator Algebra Eigenvalue Equations Extensive account of Operators Audio-visuals on-line Rigid Rotor (PowerPoint) (Good account from the Wilson Group,****) Rigid Rotor (PDF) (Good account from the Wilson Group,****) Motion in 1D QuickTime™ and a GIF decompressor are needed to see this picture. QuickTime™ and a GIF decompressor are needed to see this picture. Confined translation 1D QuickTime™ and a GIF decompressor are needed to see this picture. Free translation QuickTime™ and a Cinepak decompressor are needed to see this picture. Harmonic Oscillation 1D QuickTime™ and a Video decompressor are needed to see this picture. Vibrating diatomic molecule Thus Quantum Mechanical Rotation in 2D... quantitative derivation 1 ml ( ) expiml 2 describes a state 2 m2 l ; with precise energy E ml 0; 1; 2,.. 2I and angular momentum component in z - direction Lz ml ; ml 0; 1; 2;.... Vibrational spectroscopy Has the vibrational energy levels A vibrating diatomic molecule AB E v6 1 E ( v) 2 v5 QuickTime™ and a Video decompressor are needed to see this picture. 11 2 v4 9 2 v3 v2 v 1 A photon of energy h will be absorbed if h v0 7 2 5 2 3 2 1 2 x / 2 k mAmB mA mB Vibrational spectroscopy QuickTime™ and a Animation decompressor are needed to see this picture. QuickTime™ and a Animation decompressor are needed to see this picture. Molecules in general absorbs photons of specific frequencies (fingerprints) Motion in 2D QuickTime™ and a GIF decompressor are needed to see this picture. Confined translation 2D QuickTime™ and a Video decompressor are needed to see this picture. Rotating Diatomics QuickTime™ and a GIF decompressor are needed to see this picture. Rotation in 2D The rigid rotor QuickTime™ and a Video decompressor are needed to see this picture. Vibrating Diatomics Quantum Mechanical Rotation in 2D let us consider a particle of mass m moving in the xy plane in a circle of fixed radius a The position of the particle z is given by k r = ix + j k j a i x y where r = r r x2 y 2 a With the velocity given by dr dx dy = i + j dt dt dt v = i v x + j vy Quantum Mechanical Rotation in 2D We have It is more informtive to work in the spherical r = i x + jy coordinates (r, ) z r = i rcos + j rsin k The velocity is given by j i x r y drcos drsin v=i + j dt dt or since r is constant (r = a) d d v = - i rsin + j rcos dt dt Quantum Mechanical Rotation in 2D z k j i y r We note that |v | = vv d a2 ( )2 [sin2 cos2 ] dt d a( )2 dt v x also v r v xrx vy ry r = i rcos + j rsin dr d v = - i rsin + j rcos dt dt d a sin cos dt d a sin cos 0 dt Quantum Mechanical Rotation in 2D Let us now evaluate L = r p = m r v z k We have L = r p = ( i x + j y) ( i px j p y ) j i x y r v = i i (x px ) i j (x p y ) j i (yp x ) j j (yp y ) kxpy kyp x L m(xpy yp x )k Quantum Mechanical Rotation in 2D L m(xpy yp x )k or L z d 2 2 L [mr cos k j i x d 2 2 mr sin ]k y r dt dt v d 2 Lmr k dt L m | r | | v | k | r | | p | k r = i rcos + j rsin dr d v = - i rsin + j rcos dt dt x Quantum Mechanical Rotation in 2D The total energy is E = Ekin Epot L 2 d Epot 0 z L mr dt k k 2 d d 2 mr 2 ( )2 mr ( ) 2 p dt j dt y Ekin 2m r 2 mr 2 i v Lz 2 Ekin mr2 since I = mr 2 is moment of inertia d 2 | p | mr dt L z2 Ekin I Quantum Mechanical Rotation in 2D... qualitative origin of quantization We have L z2 Ekin I and Lz pr I mr 2 and Lz pr I mr 2 The angular momentum of a particle of mass m on a circular path of radius r in the xy-plane is represented by a vector of magnitude pr perpendicular to the plane. Quantum Mechanical Rotation in 2D... qualitative origin of quantization Consider a free paricle Its wavefunction is given by moving to the right i (x, t) AExp Et Expikx i AExp Et cos kx sinkx QuickTime™ and a GIF decompressor are needed to see this picture. AExp AExp Its momentum is given by p k The wavelength is determined from the condition (x, t) (x , t) k 2 k 2 / Etcos kx sinkx p (2 / ) h / i Etcos kx sinkx de Broglie i Quantum Mechanical Rotation in 2D... qualitative origin of quantization Consider a particle moving at a ring Its wavefunction () must satisfy ( ) ( 2) Same physical situation acceptable Not acceptable Quantum Mechanical Rotation in 2D... qualitative origin of quantization According to QM a particle of momentum p h de Broglie moves as a wave with wave length = p Standing wave must repete itself thus wavelength must be a integer fraction of the circumperence 2r n Quantum Mechanical Rotation in 2D... qualitative origin of quantization We have 2r m de Broglie h p m 0,1, 2, 3,... Boundary cond. Thus 2r h m p or hm p 2r m = 0,1,2,3,4,... Quantum Mechanical Rotation in 2D... qualitative origin of quantization hm p 2r m = 0,1, 2, 3,4,... Lz pr m The angular momentum of a particle confined to a plane can be represented by a vector of length |ml| units along the z-axis and with an orientation that indicates the direction of motion of the particle. The direction is given by the right-hand screw rule. (pr)2 h2m2 E I (2) 2 I E 2 m2 I m = 0,1, 2, 3, 4,... We might also try to solve the problem by standard procedures The Hamiltonian is again given by 2 d2 2 d H= 2 dy 2 2m dx Next making use of spherical coordinates z r 2 2 1 1 ˆ H 2 2 2 2m r r r r y i x 2 2 2 1 1 2 2 2 2 2 2 x y r r r r 2 k j Quantum Mechanical Rotation in 2D... quantitative derivation x r cos ; y = rsin Since r is a constant, a the first two derivatives with respect to r can be neglected and we have 2 2 1 ˆ H 2 2 2m r 2 2 1 ˆ H 2 2 2m r or since I = mr 2 and the Schroedinger equation is 2 2 E 2 2I Quantum Mechanical Rotation in 2D... quantitative derivation The normalized solutions are 1 ml () expiml where 2 ml 2IE here ml is just at the moment a real number or 2 2 2IE 2 2 2 m2 2IE 2 m 2 We must have that ml () ml ( 2) since (r, ) and (r, 2) represent the same point in space Quantum Mechanical Rotation in 2D... quantitative derivation ml ( 2 ) 1 expiml ( 2 ) 2 1 = exp im 2 exp im l l 2 1 = expi2ml ml ( ) 2 1 12ml ml () 2 Thus for ml ( 2 ) to be equal to ml (): 12ml 1 or ml 0, 1; 2; 3;....; Note : Similar to de Broglies condition Quantum Mechanical Rotation in 2D... quantitative derivation We have that 2IE ml Thus 2 ml2 E ; 2I ml 0; 1; 2,.. Negative m values corrspond to rotation in one direction positive m values to rotations in the other direction. For m values of the same absolute value but opposite signs the energies are the same. The two states are degenerate Quantum Mechanical Rotation in 2D... quantitative derivation 2 m2 l ; E 2I ml 0; 1; 2,.. 1 ml ( ) expiml 2 Note that the number of nodes in the real part of ml () increases with ml . The real parts of the wavefun ctions of a particle on a ring. As shorter wavelengths are achieved, the magnitude of the angul ar momentum around the z-axis grows in steps of Quantum Mechanical Rotation in 2D... quantitative derivation We have that the angular momentum in general is given by L = r p = m r v We have in the 2 - D case L = r p = ( i x + j y) ( i px j p y ) = i i (x px ) i j (x p y ) j i (yp x ) j j (yp y ) kxpy kyp x L m(xpy yp x )k L (xpy yp x )k L zk Lz xpy ypx Quantum Mechanical Rotation in 2D... quantitative derivation or quantum mechanically Lˆ z xˆ pˆ y ypˆ x Lˆ z xˆ y i y x or in spherical coordinates Lˆ z i We note that 1 ml ( ) expiml 2 is an eigenfunction to Lˆ z with eigenvalues Lz ml ; ml 0; 1; 2;....