7.1 Angular momentum Slides: Video 7.1.1 Angular momentum operators Text reference: Quantum Mechanics for Scientists and Engineers Chapter 9 introduction and Section 9.1 (first part)
Angular momentum Angular momentum operators Quantum mechanics for scientists and engineers David Miller
Angular momentum operators - preview We will have operators corresponding to angular momentum about different orthogonal axes L,, and ˆx L L ˆy ˆz though they will not commute with one another in contrast to the linear momentum operators for the different coordinate directions pˆ x, pˆ y, and pˆ z which do commute
Angular momentum operators - preview We will, however, find another useful angular momentum operator, ˆL 2 which does commute separately with each of Lˆx, Lˆy, and Lˆz The eigenfunctions for L,, and are simple ˆx L L ˆy ˆz Those for ˆL 2 the spherical harmonics, are more complicated but can be understood relatively simply and form the angular shapes of the hydrogen atom orbitals
Classical angular momentum The classical angular momentum of a small object of (vector) linear momentum p centered at a point given by the vector displacement r relative to some origin is Lrp y origin momentum p r q position of object x
Vector cross product As usual i j k CABcABsin A A A B B B x y z x y z i( A B A B ) j( A B A B ) k( A B A B ) y z z y x z z x x y y x where i, j, and k are unit vectors in x, y, and z directions and A x is the component of A in the x direction and similarly for the y and z directions and the components of B
Vector cross product In i j k CABcABsin A A A B B B x y z x y z i( A B A B ) j( A B A B ) k( A B A B ) y z z y x z z x x y y x C is perpendicular to the plane of A and B just as the z axis is perpendicular to the plane containing the x and y axes in right-handed axes is the angle between the vectors A and B c is a unit vector in the direction of the vector C
Vector cross product Note that, in i j k CABcABsin A A A B B B x y z x y z i( A B A B ) j( A B A B ) k( A B A B ) y z z y x z z x x y y x the ordering of the multiplications in the second line is chosen to work also for operators instead of numbers for one or other vector the sequence of multiplications in each term is always in the sequence of the rows from top to bottom
Angular momentum operators With classical angular momentum L rp we can explicitly write out the various components Lx ypz zpy Ly zpx xpz Lz xpy ypx Now we can propose a quantum mechanical angular momentum operator ˆL based on substituting the position and momentum operators Lˆ rˆpˆ ir and similarly write out component operators
Angular momentum operators Analogously, we obtain three operators ˆ L ˆˆ ˆˆ x ypz zpy i y z z ˆ L ˆˆ ˆˆ y zpx xpz i z x x y z ˆ L ˆˆ ˆˆ z xpy ypx ix y y x which are each Hermitian and so, correspondingly, is the operator ˆL itself
Commutation relations The operators corresponding to individual coordinate directions obey commutation relations LL ˆ ˆ ˆ ˆ ˆ, ˆ ˆ x y LL y x Lx L y ilz LL ˆ ˆ ˆ ˆ ˆ, ˆ ˆ y z LL z y Ly L z ilx LL ˆ ˆ ˆ ˆ ˆ ˆ ˆ z x LL x z Lz, L x ily These individual commutation relations can be written in a more compact form Lˆ Lˆ ilˆ
Commutation relations Unlike operators for position and for linear momentum the different components of this angular momentum operator do not commute with one another Though a particle can have simultaneously a welldefined position in both the x and y directions or have simultaneously a well-defined momentum in both the x and y directions a particle cannot in general simultaneously have a welldefined angular momentum component in more than one direction
7.1 Angular momentum Slides: Video 7.1.3 Angular momentum eigenfunctions Text reference: Quantum Mechanics for Scientists and Engineers Section 9.1 (remainder)
Angular momentum Angular momentum eigenfunctions Quantum mechanics for scientists and engineers David Miller
Spherical polar coordinates z The relation between spherical polar and Cartesian coordinates is (x, y, z) x r sincos y rsinsin z r cos q f r y is the polar angle, and is the azimuthal angle x
Spherical polar coordinates z In inverse form 2 2 2 r x y z (x, y, z) 2 2 1 x y sin 2 2 2 x y z q f r y 1 y tan x x
Angular momentum in spherical polar coordinates With these definitions of spherical polar coordinates and with standard partial derivative relations of the form r x x r x x for each of the Cartesian coordinate directions we can rewrite the angular momentum operator components in spherical polar coordinates
Angular momentum in spherical polar coordinates From LL ˆ ˆ ˆ ˆ ˆ ˆ ˆ x y LL y x Lx, L y ilz LL ˆ ˆ LL ˆ ˆ Lˆ, Lˆ i Lˆ and LL ˆ ˆ ˆ ˆ ˆ ˆ ˆ z x LL x z Lz, L x ily we obtain ˆ Lx isin cotcos ˆ Ly icos cotsin Lˆz i y z z y y z x
L z eigenfunctions and eigenvalues Using Lˆz i we solve for the eigenfunctions and eigenvalues of Lˆz The eigen equation is Lˆz m where m is the eigenvalue to be determined The solution of this equation is expim
L z eigenfunctions and eigenvalues The requirements that the wavefunction and its derivative are continuous when we return to where we started i.e., for 2 mean that m must be an integer positive or negative or zero Hence we find that the angular momentum around the z axis is quantized with units of angular momentum of