Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.
In classical physics if we have two distinct physical states ψ1 6=ψ2 of a physical system and we know that ψ1ORψ2 is a true statement we can easily deduce that ψ1XORψ2 is a true statement too. In Quantum Mechanics however we encounter a novel possibility for quantum coherent superposition. It has been verified in numerous experiments that a qubit can be prepared in a linear combination of two orthogonal states, and this parallel existence in the quantum realm, in the form ψ1ANDψ2, is what requires caution when we draw conclusions from a given set of premises — the truth of ψ1ORψ2 now does not lead to the truth of ψ1XORψ2. If a qubit at point x is in a state ψ1XORψ2 then ψ1 and ψ2 are called distinguishable states. Logically, if the qubit at point x is in a state ψ1XNORψ2 the two states ψ1 and ψ2 will be indistinguishable. From the requirement for mathematical consistency it follows that two states ψ1 and ψ2 cannot be both distinguishable and indistinguishable at the same time. The concept of distinguishability is intimately tied with the notion of quantum complementarity. While the quantum amplitudes evolve linearly according to the Schrodinger equation, the physical observables are obtained from the underlying quantum amplitudes through nonlinearity prescribed by Born’s rule. Thus if quantum states ψ1(x) 6=0 and ψ2(x) 6=0 are indistinguishable at a point x (coherent superposition), that is ψ1(x)ANDψ2(x), the probability distribution (observed intensity) is given by P = |ψ1(x) + ψ2(x)|. The density matrix of the setup is a pure type one, ρ= ( |ψ1| 2 ψ1ψ ∗ 2
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.