Basic Quantum Mechanics


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1 Basic Quantum Mechanics Postulates of QM  The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn)  To every observable (physical magnitude) there corresponds an Hermitian operator given by the following rules: a) The operator corresponding to the Cartesian position coordinate x is x. Similarly for y and z b) The operator corresponding to the x component of the linear momentum p x is ( h )( i x ) c) to obtain the operator corresponding to any other observable first write down the classical expression of the observable in terms of x, y, z, p x, p y, p z and then replace each quantity by its corresponding operator according rules and. 3 The only possible result which can be obtained when a measurement is made of an observable whose operator is A is an eigenvalue of A. 4 Let α be an observable whose operator A has a set of eigenfunctions Φ with corresponding eigenvalues a i. If a large number of measurements of α are made on a system in the state Ψ the value obtained is given by A = Ψ A Ψ dτ where the integral is taken over all space. 5 If the result of a measurement of α is a r, corresponding to the eigenfunction Φ r, then the state function after the measurement is Φ r 6 The time variation of the state function of a system is given by: Ψ = HΨ t i Where H is the Hammiltonian of the system Probability of result of a measurement Let us suppose a set of eigenvalues a of an operator A are discrete, and that the state function Ψ and all the eigenfunctions Φ are normalized. To find the probability p r that the result of the measurement of an observable α is a particular value a r we expand the wavefunction in terms of the eigenfunctions:
2 Ψ= c Φ Then the probability will be The coefficient c r is obtained as p r = c r c = Φ Ψ dτ If the coefficients c are known, an expression for the expectation value is A = p a = c a r Continuous eigenvalues. Let γ be an observable whose operator G has eigenvalues k which form a continuous spectrum, i.e. any real value k is a possible result of a measurement of γ. For simplicity we will consider only the one dimensional case for a system consisting of a single particle. If Φ(k,x) are the eigenfunctions of G the wavefunction Ψ can be written as ( ) ( ) (, ) Ψ x = g k Φ k x dk where the significance of the function g(k) is similar to the coefficients c. Specifically, g( k) dk is the probability that if the observable γ is measured, the value obtained lies in the range k k+ dk. Linear momentum An important example of an observable with continuous eigenvalues is the linear ikx Φ kx, = ce with eigenvalues momentum. The eigenfunctions of the operator p x are ( ) k. For the linear momentum the wave function can be expressed as ( ) ( ) ikx Ψ x = c g k e dk The functions g( k) ( x) ikx ( ) Ψ( ) g k x e dx Ψ are related through the Fourier transform. the proportionality constant can be obtained from the condition that ( ) dk = g k a) Time variation of a wave function
3 Suppose a system with a Hammiltonian that does not vary with time. If the wavefunction at a time t=0 is known the function at a later time is obtained as ie t Ψ () t = c u exp where u is the eigenvalue of H with energy E. Assuming that the values and the coefficients c are given by the relation ( 0) c = u Ψ dx IF the eigenvalues of the Hammiltonian have a continuous spectrum with corresponding eigenfunctions u k the wavefunction is expressed as ie (, ) ( ) exp kt Ψ x t = g k u k dk where ( ) = Ψ(,0) g k u x dx k Time variation of expectation value of an observable The time variation of the expectation value of an observable with operator A for a system with a wavefunction Ψ is = Ψ ( ) Ψ = t i i A AH HA dτ AH HA Schodinger equation One particle in one dimension moving in a potential V(x) will have a Hammiltonian function: px H = Ek + V ( x) = + V( x) m where p is the linear momentum of the particle. The first term in the kinetic energy and the second term is the potential energy. To find the expression of the Hammiltonian replace the impulse by ( i ) d dx H d = + V mdx
4 If u(x) is an eigenfunction of the Hammiltonian with eigenvalue E (this is the energy of the system), we can write Hu x ( ) = Eu( x) d mdx Or + V u( x) = E u( x) thus ( ) d u x m E V u x dx ( ) ( ) + = 0 This is the Schodinger equation in one dimension. In three dimensions H ( px + py + pz ) = + V ( x, y, z) and u( xyz) ( E V) u( xyz) m m,, +,, = 0 Parity Let us consider a one dimensional function f(x). If f ( x) = f ( x) the function is said to have even parity. If f ( x) = f ( x) the function has odd parity. A function that does not satisfy either of these two conditions is said to have mixed parity. A mixed parity function can always be expressed as a sum of two functions, one with even parity and the other with odd parity. Continuity The eigenfunction u is continuous everywhere. The derivatives of u are also continuous everywhere, except where the potential function has an infinite discontinuity (only possible in theory, cannot happen in an actual physical situation) Commuting operators In classical physics, we can know both the position and the momentum with no limitation. This is not the case on QM. Two observables can be known simultaneously only if a measurement doesn t change the state of the system. Otherwise, when measuring one observable it will change the eigenfunction of the other observable. This is the case if the wavefunction Ψ is an eigenfunction of both operators. Consider operators A and B. We measure the observable corresponding to A and then that corresponding to B. Let us assume that Ψ(x) is an eigenfunction of A and B. ( ) α ( ) α β ( ) B AΨ n x = n BΨ n x = n n Ψn x But it is also true that A BΨ ( x) = β AΨ ( x) = α β Ψ ( x) n n n n n n
5 So the system is unchanged by the measurement the wavefunction is an eigenfunction of both operators! In this case A BΨ n( x) = B AΨn( x). Two operators that have a common set of eigenfunctions are said to commute. For an arbitrary function if B A ( x) A B ( x) [ A B] ( x) Θn Θn, Θ n = 0 it is said that the commutator is zero and the two operators commute. Orbital angular momentum The operator for the components of the orbital momentum are Lx Ly L z. For the square of this magnitude the operator is L = Lx + Ly + Lz The operator L commutes with L x, L y, L z. The component operators do not commute with each other. The commutation rule for the angular momentum operators is L x, L i L y = z and cyclic permutations. The common eigenfunctions of L and L z are the spherical harmonics Y. The eigenvalues of L and L z are given by: ( ) LY l l Y L Y = m Y = + z Where l 0 and l m l The spherical harmonics are functions of the polar angles θ φ. They are orthogonal and normalized π π 0 0 π π 0 0 ' ' ' ' ( ) Y Y sinθ dθ dφ = l ' = l and m' = m ( ) Y Y sinθ dθ dφ = 0 otherwise The spherical harmonics l=0,, are Y00 = 4π 3 3 Y0 = cosθ Y ± = sinθ exp( ± iφ) 4π 8π Y0 = ( 3cos θ ) Y ± = cosθ sinθ exp( ± iφ) Y ± = sin θ exp( ± iφ) 6π 8π 3π
6 Ladder operators Ladder operators are defined as combination of L x and L y L = L + il L = L il + x y x y They satisfy the relations + = + + l, m+ { ( ) ( ) } L Y l l m m Y { ( ) ( ) } L Y = l l+ m m Y l, m If we consider the set of eigenfuctions as arranged on a ladder, the operator L + converts the function Y into the function one rung up, and L  into a function one rung down. This is the origin of the name ladder operators. They are also called raising operator and lowering operator. Ladder operators also fulfill L Y = L Y, = 0 + ll l l
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