The time-independent Schrödinger equation is given by

Size: px

Start display at page:

Download "The time-independent Schrödinger equation is given by"

August Hancock
7 years ago
Views:

1 4 Operators The time-independent Schrödinger equation is given by ] [ 2 2m 2 + V (r) ψ(r) =Ĥψ(r) =Eψ(r) (140) The terms in the brackets are called the Hamiltonian and in quantum mechanics is an operator. An operator transform one function into another function. One example is the derivative operator ˆDf(x) =f (x) (141) We now consider some rules of operators. The sum, differences and product of two operators Â and ˆB is given by (Â + ˆB)f(x) = Âf(x)+ ˆBf(x) (142) (Â ˆB)f(x) = Âf(x) ˆBf(x) (143) and obey the associate law of multiplication Â ˆBf(x) = Â[ ˆBf(x)] (144) Â( ˆBĈ) = (Â ˆB)Ĉ (145) Lets consider an example where Â = d/dx and ˆB =ˆx. First we evaluate Â ˆB second lets evaluate ˆBÂ Â ˆB = d dx [xf(x)] = f(x)+xf (x) = (1 + ˆx ˆD)f(x) (146) d ˆBÂf(x) = ˆx[ dx f(x)] = xf (x) (147) thus we see that for operators Â ˆB is not always ˆBÂ. define the commutator of two operators as It is convenient to [Â, ˆB] =Â ˆB ˆBÂ (148) if Â ˆB = ˆBÂ then the commutator is zero and the two operators are said to commute. Examples [ 3, d ] =3 d dx dx d dx 3 = 0 (149) 21

2 and [ ] d dx,x = ˆDx x ˆD = 1 (150) Lets consider the commutator between the position operator x and the momentum operator ˆp i d/dx [ˆp, x] = ˆpx xˆp = i ( d dx x x d )= i (151) dx The square of an operator is defined as and the exponential of an operator as 4.1 Linear operators Â 2 = ÂÂ (152) Â2 exp(â) = 1 + Â + 2! + Â3 + (153) 3! An important class of operators are linear operators. An operator Â is said to be linear if and only if it has the following two properties and Â[f(x)+g(x)] = Âf(x)+Âg(x) (154) Â[cf(x)] = câf(x) (155) where c is a constant and f(x) and g(x) are functions. We ll consider two examples first is the ˆD and the second is Â2 = () 2. For the differential operator we see that (d/dx)[f(x) +g(x)] = (d/dx)f(x) + (d/dx)g(x) (156) (d/dx)[cf(x)] = c(d/dx)f(x) (157) and therefore the operator is linear. What about the square operator Â 2 [f(x)+g(x)] = (f(x)+g(x)) 2 Â2 f(x)+â2 g(x) (158) and, thus, is nonlinear. It turns out that almost every operators in quantum mechanics are linear operators. 22

3 4.2 Eigenfunctions and eigenvalues An eigenfunction of an operator Â is a function which when the operator works on it return the functions times a constant Âf(x) =af(x) (159) where a is called the eigenvalue. As an example, what would be an eigenfunction of the differentiation operator? Either cosine, sine or exponentials (d/dx)(f x) =(d/dx) exp( ax) = a exp(ax) = af(x) (160) Another example would be the Schrödinger equations for a particle in a 1D box d 2 f(x) + 2m dx 2 E xf(x) = 0 (161) 2 which we have already explored. 4.3 Operators in quantum mechanics In quantum mechanics, physical observables (e.g., energy, momentum, position, etc.) are represented mathematically by operators. For instance, the operator corresponding to energy is the Hamiltonian operator ] [ 2 2m 2 + V (r) ψ(r) =Ĥψ(r) =Eψ(r) (162) The operators are found be writing down the classical expression for the property of interest and then substitute x ˆx = x (163) p x =ˆp = i x (164) The classical expression for the Hamiltonian of a system (which is just a reformulation of Newton s equations) is using the substitution we get H = T + V = px2 + V (x) (165) 2m d 2 Ĥ = ˆT + ˆV = 2 2m dx + ˆV (x) (166) 2 23

4 and as we see that all these operators are linear. The properties of a system is related to the eigenvalues of the operator Ĥψ i = E i ψ i (167) where the energy E i is the eigenvalues of the Hamiltonian. Quantum mechanics postulates that a measurement of a property A most give one of the eigenvalues of the operator Â. As an example consider a stationary state Ψ(x, t) = exp( iet/ )ψ(x) (168) operating on this wavefunction with the Hamiltonian gives ĤΨ(x, t) = exp( iet/ )Ĥ = exp( iet/ )E exp( iet/ ) =EΨ(x, t) (169) and, thus, stationary states are eigenfunctions of the Hamiltonian. 4.4 Degeneray If we have n independent wavefuctions each having the same eigenvalue a Ĥψ i = aψ i i =1, 2, 3, (170) The every linear combination of these wavefunctions φ = i c i ψ i (171) will also be an eigenfunction of the operator with the same eigenvalues since Ĥφ = Ĥ i c i ψ i = i c i Ĥψ i = i c i aψ i = a i c i ψ i = aφ (172) 4.5 Expectation values of an operator In quantum mechanics the average value (or expectation value) of an property is given by A = ψ (r)âφ(r)dr (173) if the wave function is normalized otherwise ψ A = (r)âφ(r)dr ψ (r)φ(r)dr (174) 24

5 Thus is a system is in an eigenstate of the operator Â, i.e. Âψ = aψ (175) where ψ is assumed normalized, then the expectation value is given by A = ψ (r)âφ(r)dr (176) = ψ (r)aφ(r)dr (177) = a ψ (r)φ(r)dr (178) = a (179) 25

1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators