Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.1 OPERATOR FORMALISM spectrum of the system is continuous. This system has discrete negative and continuous positive eigenval ...

QUANTUM OPERATORS IfA|an〉=a|an〉for allN 1 ≤n≤N 2 ,then |ψ〉= ∑N^2 n=N 1 dn|an〉satisfiesA|ψ〉=a|ψ〉for any set ofdi. For a general s ...

19.1 OPERATOR FORMALISM whilstthatforBA|ψ〉is simply x ∂ψ ∂x , which is not the same. If the result AB|ψ〉=BA|ψ〉 is true forallket ...

QUANTUM OPERATORS Simple identities amongst commutators include the following: [A, B+C]=[A, B]+[A, C], (19.17) [A+B, C]=[A, C]+[ ...

19.1 OPERATOR FORMALISM defining series, we have f(A)=2 ∑∞ m=0 (−1)mA^2 m+1 (2m+1)! ∑∞ n=0 (−1)nA^2 n (2n)! . Writingm+nasrand r ...

QUANTUM OPERATORS later algebraic convenience: [ A, eλB ] = [ A, ∑∞ n=0 (λB)n n! ] = ∑∞ n=0 λn n! [A, Bn] = ∑∞ n=0 λn n! nBn−^1 ...

19.2 PHYSICAL EXAMPLES OF OPERATORS quantum-mechanical operators are those corresponding to positionrand mo- mentump. One prescr ...

QUANTUM OPERATORS RHS gives (−i
)^2 2 m ∂ ∂x ∂ ∂x + (−i
)^2 2 m ∂ ∂y ∂ ∂y + (−i
)^2 2 m ∂ ∂z ∂ ∂z . The potential energyV, being ...

19.2 PHYSICAL EXAMPLES OF OPERATORS Consider first LxLy=−
^2 ( y ∂ ∂z −z ∂ ∂y )( z ∂ ∂x −x ∂ ∂z ) =−
^2 ( y ∂ ∂x +yz ∂^2 ∂z∂x −y ...

QUANTUM OPERATORS with [ L^2 ,Lz ] = [ L^2 x+L^2 y+L^2 z,Lz ] =Lx[Lx,Lz]+[Lx,Lz]Lx +Ly [ Ly,Lz ] + [ Ly,Lz ] Ly+ [ L^2 z,Lz ] =L ...

19.2 PHYSICAL EXAMPLES OF OPERATORS Consider firstL^2 |ψ′〉, recalling thatL^2 commutes with bothLxandLyand hence withU: L^2 |ψ′〉 ...

QUANTUM OPERATORS operate repeatedly on it with the (down) ladder operatorD, we will generate a state|ψd〉which, whilst still an ...

19.2 PHYSICAL EXAMPLES OF OPERATORS other set of four operators with the same commutation structure would result in the same eig ...

QUANTUM OPERATORS 19.2.2 Uncertainty principles The next topic we explore is the quantitative consequences of a non-zero com- mu ...

19.2 PHYSICAL EXAMPLES OF OPERATORS In the second line, we have moved expectation values, which are purely numbers, out of the i ...

QUANTUM OPERATORS hence formally an eigenstate ofL^2 with= 0, all three components of angular momentum could be (and are) known ...

19.2 PHYSICAL EXAMPLES OF OPERATORS 19.2.3 Annihilation and creation operators As a final illustration of the use of operator me ...

QUANTUM OPERATORS an arbitrary complete set of orthonormal base states|φi〉and using equation (19.11), is as follows: 〈ψ|B^2 |ψ〉= ...

19.2 PHYSICAL EXAMPLES OF OPERATORS In a similar way it can be shown thatA†parallels the operatorUof our angular momentum discus ...

QUANTUM OPERATORS The proof, which makes repeated use of [ A, A† ] =
, is as follows: Am(A†)m=Am−^1 AA†(A†)m−^1 =Am−^1 (A†A+
)(A ...