Survival Facts from Quantum Mechanics

Transcription

1 Survival Facts from Quantum Mechanics Operators, Eigenvalues an Eigenfunctions An operator O may be thought as something that operates on a function to prouce another function. We enote operators with hats, but also use hats to inicate unit vectors like xˆ, ŷ an ẑ. The context shoul make it clear which is which): Ôf x) = gx) In most cases, the operators of quantum mechanics are linear. Operators are linear if they have properties: Ô[ f x) + gx)] = Ôf x) + Ôgx) Examples: Linear operators: Ôcf x) = côf x) where c is a constant c can be a complex number: c = a + ib, i = 1 ) x multiplication by x): x[ f x) + gx)] = xf x) + xgx) x ifferentiation with respect to x): x [ f x) + gx)] = x f x) + x gx) A nonlinear operator: square root operator): f x) + gx) f x) + gx) The eigenvalues an eigenfunctions of an operator A are those numbers a j an functions ϕ j which satisfy  j = a j j where j is just a label for the various eigenfunctions an corresponing eigenvalues which satisfy this equation. In other wors, when A operates on one of its eigenfunctions, say ϕ 3, the result is a 3 ϕ 3 - just ϕ 3 back again, multiplie by the eigenvalue a 3. Note that if we multiply an eigenfunction of a linear operator by a constant c we still have an eigenfunction:  j = a j j Âc j ) = câ j = ca j j ) = a j c j ) 1

2 so that an eigenfunction ϕ j an the function χ j =cϕ j are not consiere as inepenent eigenfunctions. i.e., Since any eigenfunction is still an eigenfunction when multiplie by a constant, eigenfunctions which iffer only by a multiplicative constant are not consiere to be istinct.) Examples: 1) The ifferential operator {/x} has an infinite set of eigenfunctions {e kx } with eigenvalues k: where k may take on any value. x ekx = ke x { } : 2) The operator x # x also has an infinite set of eigenfunctions x n ; n 0 x x n = xnx n1 ) = nx n # x This example allows us to emonstrate that a linear combination of eigenfunctions is not an eigenfunction unless the two eigenfunctions have the same eigenvalue). For example, there is no number c that satisfies the equation: x # x x2 + x 3 ) = c x 2 + x 3 ) 3) The operator { 2 /x 2 } has a set eigenfunctions of the form {cos kx; k = any real number} an k 2 is the eigenvalue: 2 x 2 [cos kx] = x [k sin kx] = k 2 [cos kx] Note that the set of functions {sin kx; k = any real number} are also eigenfunctions with the same eigenvalue: 2 [sin kx] = # x 2 x [k cos kx] = k 2 [sin kx] Therefore, for any given value of k, cos kx, an sin kx are eigenfunctions of { 2 /x 2 } with the same eigenvalue k 2. This means that any combination of cos kx an sin kx is also an eigenfunction. 2 [a cos kx + bsin kx] = k 2 [acos kx + bsin kx] # x 2 In particular, if a = 1 an b = i = 1 we have 2 [cos kx + isin kx] = 2 [e ikx ] = k 2 [e ikx ] # x 2 # x 2 so that {e ikx ; k = any real number} is an alternative set of eigenfunctions of { 2 /x 2 }. 2

3 Commutators The commutator of two operators A an B is efine as Â, ˆB # = Â ˆB ˆBÂ if [A,B ] = 0, then A an B are sai to commute. In general, quantum mechanical operators cannot be assume to commute. Examples: When evaluating the commutator for two operators, it is sometimes helpful to keep track of things by operating the commutator on an arbitrary function, fx). 1) Evaluate x, # x : = x f x) x x, # x f x) = x x ) x x +, - f x) f x) f x) xf x)) = x x f x) x x x x x = f x) x, # x f x) = f x). x, # x = 1 We see that the effect of operating on any arbitrary fx) with [x, /x] is to prouce fx), so that the last equation is generally true. 2) In quantum mechanics, the operator for linear momentum in the x irection is ˆp x = i x where = h/2π). Lets evaluate [x, p x ]: x x # x x [ x, ˆp x ] = xˆp x ˆp x x = i = x, + i ) x, - = i = i 3) In classical mechanics, the angular momentum of a particle aroun the origin is a vector quantity, L, which is efine as L = r p. The angular momentum of a charge particle like an electron) is relate to its magnetic moment, m, as shown in the illustration below. L = ˆx ŷ ẑ x y z = p x p y p z yp z zp y )ˆx + zp x xp z )ŷ + xp y yp x )ẑ so, ientifying the components of L, L x = yp z zp y ; L y = zp x xp z ; L z = xp y yp x 3

4 L x, L y, an L z are the components of a single particle s angular momentum. To get the quantum mechanical operators L =L xxˆ + L yŷ + L zẑ) we insert the quantum mechanical operators for the linear momenta, p u = /i) / u u = x, y, z), to obtain: ˆL x = # y i z z y ; ˆLy = # z i x x z ; ˆLz = # x i y y x The reaer shoul use these expressions to operate on an arbitrary function fx, y, z) to evaluate the commutators [ ˆL x, ˆL y ] = i ˆL z ; [ ˆL y, ˆL z ] = i ˆL x ; [ ˆL z, ˆL x ] = i ˆL y Quantum Mechanical Operators an Wavefunctions If ϕ is to be consiere a well behave function, then we eman that it have the following properties: a) ϕ must be continuous no breaks) b) ϕ must have continuous erivatives no kinks) c) ϕ must be normalizable. To be normalizable, ϕ must obey the conition the symbol τ symbolizes integration over all space): # = C, where C is a finite constant. Then we can normalize if we multiply by 1 C : # # C C ) = 1. Of central importance is the time-inepenent Schröinger equation H = E. where H is the Hamiltonian or energy) operator for the system. Ψ is calle the wavefunction or state function for the system an must be well behave in the sense inicate above. In quantum mechanics, physically observable quantities are associate with Hermitian operators eg., the energy of the system, the momentum of the system or of particles within the system, the position of particles in the system, etc.) If an operator A is Hermitian it has the property for all well-behave functions ϕ an χ. Â# = # Â), If a system is in a state escribe by Ψ, then the expectation value we woul observe for the property associate with the Hermitian operator A is given by  # Â, if it is a physically observable quantity, it must real: 4

5  =  so #  = # Â). Thus, the Hermitian property of operators associate with observables guarantees that calculate quantities for these observables will be real. Example: Is p x Hermitian? ˆp x x = # # Integrate by parts, = i # # i # i x ) + x = i # uv = uv # vu # ) # x # ) + x = # i x ) + x = x ) + x = ˆp x ) x Note that if p x int have the factor of i in front, it woulnt be Hermitian. Orthogonality Definition): Two functions an are sai to be orthogonal if # = 0 Important property of Hermitian Operators: Eigenfunctions of a Hermitian operator are orthogonal. In the case where two or more eigenfunctions have the same eigenvalue, then the eigenfunctions can be mae to be orthogonal). proof: suppose ϕ i an ϕ j are eigenfunctions of A with respective eigenvalues a i an a j such that a i a j :  i = a i i ;  j = a j j # By use of the Hermitian property we get: # i  j = # j  i ), now operate with A on both sies of the equation: a j # i j = a i # j i. a i is real because A is Hermitian. Then we have a i a j ) j # i = 0. j # i = 0. Now, if a = a, then we are free to combine ϕ an ϕ i an we will still have an j i j eigenfunction see the example concerning 2 /x 2 above). For example, suppose ϕ an 1 ϕ 2 both have eigenvalue a with respect to operator A. Then we can take 1 = 1 an 2 = 2 + c 1 an we can choose c = # 1 2 # 1 1. You can show that χ 1 an χ 2 are orthogonal an both still have eigenvalue a. QED. 5

6 Very Important Fact: Commuting operators have common eigenfunctions. Suppose A an B commute: [A, B ] = A B B A = 0. Let {ϕ i } be the set of eigenfunctions of A : then, ˆB  i  i = a i i ) = ˆB a i i ) = a ˆBi i ) ) = a i ˆB i ) but  ˆB = ˆB so  ˆB i the function B ϕ i is an eigenfunction of A with eigenvalue a i. If ϕ i is the only eigenfunction of A with eigenvalue a i, then B ϕ i ϕ i in other wors, B ϕ i can only be an eigenfunction of A with eigenvalue a i if it iffers from ϕ i by a constant multiplicative factor p. 2 of this hanout). Thus, ˆB i = b i i for some constant b i. If a i is a egenerate eigenvalue, i.e. there are more than one eigenfunctions of A with eigenvalue a i, then we can take linear combinations of these eigenfunctions to satisfy this conition. 6