Physics
Hermitian Operator
A Hermitian operator is a fundamental concept in quantum mechanics. It is an operator that is equal to its own adjoint, meaning that its matrix is equal to its complex conjugate transpose. Hermitian operators have real eigenvalues and play a crucial role in the formulation of quantum mechanics, particularly in the context of observables and measurements.
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7 Key excerpts on "Hermitian Operator"
eBook - PDF- Michael A. Parker(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
The eigenvalues are real. Physical systems need the real eigenvalues so that the results of measurement will yield real results. 4.9.1 Adjoint, Self-Adjoint and Hermitian Operators Let ^ T : V ! V be a linear transformation (Figure 4.9.1) defined on a Hilbert space V ¼ Sp fj n i : n ¼ 1 , 2 , . . . g . Let j f i , j g i be two elements in the Hilbert space. We define the adjoint operator ^ T þ to be the operator that satisfies g ^ Tf D E ¼ ^ T þ g f D E ð 4 : 9 : 1 Þ An operator ^ T is self-adjoint or Hermitian if ^ T þ ¼ ^ T . Previous sections define the adjoint ^ T þ as connected with the dual vectors space. We can demonstrate Equation 4.9.1 using the previous definition of the adjoint. Using the notation ^ T j f i ¼ j ^ Tf i , we find g ^ Tf D E ¼ g ^ T f ¼ ^ T þ g h i þ f ¼ ^ T þ g E h i þ f ¼ ^ T þ g f D E Example 4.9.1 If ^ T ¼ @=@ x then find ^ T þ for the following Hilbert space HS ¼ f : @ f x ð Þ @ x exists and f ! 0 as x ! 1 & ' Solution We want ^ T þ such that h f j ^ Tg i ¼ h ^ T þ f j g i . Start with the quantity on the left f ^ Tg D E ¼ Z 1 1 dx f x ð Þ ^ Tg x ð Þ ¼ Z 1 1 dx f x ð Þ @ @ x g x ð Þ 234 Physics of Optoelectronics The procedure usually starts with integration by parts: f ^ Tg D E ¼ f x ð Þ g x ð Þ 1 1 Z 1 1 dx @ f x ð Þ @ x g x ð Þ In most cases, the surface term produces zero. Notice the Hermitian property of the operators depends on the properties of the Hilbert space. In the present case, the Hilbert space is defined such that f ð1Þ g ð1Þ f ð1Þ g ð1Þ ¼ 0; most physically sensible func-tions drop to zero for very large distances. Next move the minus sign and partial derivative under the complex conjugate to find f ^ Tg D E ¼ Z 1 1 dx @ f x ð Þ @ x ! g x ð Þ ¼ ^ T þ f g D E Note everything inside the bra h j must be placed under the complex conjugate ð Þ in the integral.
eBook - PDF- Mackillo Kira, Stephan W. Koch(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
At the same time, the conjugation of a sum of operators can be performed individually while one must reverse the order of operators in conjugated operator products. Even though Eq. (5.3) shows that ˆ p † = ˆ p, this is not valid for a generic operator ˆ F † = ˆ F . In the special case of ˆ F † = ˆ F , the operator ˆ F is denoted as self-adjoint or Hermitian. Hermitian Operators have a special importance in measurements because they construct the very important class of operators that define observables, related to experimentally measurable properties such as position, intensity, momentum, etc. Since measurements must produce real-valued outcomes, expectation values corresponding to observables must also be real valued even though the wave functions are generally complex. One can easily convince oneself that any Hermitian Operator with ˆ F † = ˆ F , produces strictly real-valued expectation values – and vice versa – based on Eqs. (3.17) and (5.4) for arbitrary wave functions, see Exercise 5.1. Thus, any observable physical quantity must be represented by a Hermitian Operator. 5.2 Eigenvalue problems The treatment of Hermitian Operators ˆ O becomes particularly efficient if we consider the eigenvalue problem O (r) φ λ (r) = λ φ λ (r), (5.6) 88 Central concepts in measurement theory where λ denotes a generic eigenvalue. Due to the diagonality of O (r), Eq. (5.6) is not the most general form, however, it already covers all the examples in Chapters 3–4. For example, the stationary Schrödinger equation (4.1) can be written in this format. Mathematically, Eq. (5.6) defines a linear problem and we may apply several powerful relations known from linear algebra. To explore the implications of quantum mechanics, one may view Eq. (5.6) as a typical matrix eigenvalue problem. However, the dimension of the matrices may approach infinity if all quantum-mechanically possible cases are included.
eBook - PDFQuantum Mechanics I
A Problem Text
- David DeBruyne, Larry Sorensen(Authors)
- 2018(Publication Date)
- Sciendo(Publisher)
The | ψ i > are the eigenvectors/eigenstates of the system. The general state vector is the superposition or linear combination of all eigenstates, i.e. , | ψ> = c 1 | ψ 1 > + c 2 | ψ 2 > + c 3 | ψ 3 > + · · · = ∞ summationdisplay i =1 c i | ψ i > in the case of an infinity of eigenstates. Each of the coefficients c i are scalars that indicate the relative “amount” of eigenstate | ψ i > in the superposition that and are often called probability amplitudes because the probability of measuring the corresponding eigenvalue is often | c i | 2 . A coefficient can be zero meaning that the corresponding eigenstate is absent from that state vector. Observable quantities are those that can be physically measured. That all Hermitian Operators can be diagonalized was employed in the first chapter to show that the eigenvalues of Hermitian Operators are real numbers. Numbers used to describe physical quantities are necessarily real numbers. This fact is essential to the observables postulate. Also, the eigenvectors of Hermitian Operators are orthogonal, therefore, the eigenvectors can be made orthonormal. Further, the eigenvectors of Hermitian Operators form a basis appropriate to the space. These four properties of Hermitian Operators will be further supported in chapter 3 following additional development of unitary transformations. The observable quantities of position, momentum, energy, and angular momentum are focal. Though entwined within many of the postulates, quantum mechanics requires the field of complex numbers. The observables postulate specifically denotes Hermitian Operators. Hermitian means A = A † . The dagger means adjoint or transpose conjugate, thus the elements are implied to be from the field of complex numbers. The eigenvalue postulate is definitely non-classical. The only possible result of a measure-ment is an eigenvalue of the operator representing the physical quantity being measured.
eBook - PDF- Michael T. Vaughn(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
2.2 Linear Operators 57 Definition 2.9. The operator A is self-adjoint , or Hermitian , if A † = A (2.92) In terms of the operator matrix elements, this requires ( y, A x ) = ( x, A y ) ∗ (2.93) for every pair of vectors x, y . Remark. It follows that the diagonal matrix elements ( x, A x ) of a Hermitian Operator A are real for every vector x . In a complex vector space, it is also sufficient for A to be Hermitian that ( x, A x ) be real for every vector x , but not so in a real vector space, where ( x, A x ) is real for all x for any linear operator A (see Exercise 2.7). Remark. Self-adjoint is defined by Eq. (2.92) and Hermitian by Eq. (2.93). In a finite-dimensional space, these two conditions are equivalent and the terms self-adjoint and Hermit-ian are often used interchangeably. In an infinite-dimensional space, some care is required, since the operators A and A † may not be defined on the same domain. Such subtle points will be discussed further in Chapter 7. 2.2.4 Change of Basis; Rotations; Unitary Operators The coordinates of a vector x , and elements of the matrix representing a linear operator A , depend on the basis chosen in the vector space V . Suppose x 1 , x 2 , . . . and y 1 , y 2 , . . . are two sets of basis vectors in V . Then define a linear operator S by S x k = y k = j S jk x j (2.94) ( k = 1 , 2 , . . . ). S is nonsingular (show this), and x k = S − 1 y k ≡ j S jk y j (2.95) where the S jk = ( S − 1 ) jk are the elements of the matrix inverse of the matrix ( S jk ) . The coordinates of a vector x = k ξ k x k = η y (2.96) in the two bases are related by ξ k = S k η η = k S k ξ k (2.97) The operator S can also be viewed as a transformation of vectors: x ≡ S x (2.98) 58 2 Finite-Dimensional Vector Spaces in which x is a vector whose coordinates in the basis y 1 , y 2 , . . . are the same as those of the original vector x in the basis x 1 , x 2 , .
eBook - PDFFunctional Analysis for Physics and Engineering
An Introduction
- Hiroyuki Shima(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
The analogy holds only when the operator is self-adjoint. Keypoint: Spectrum-eigenvalue correspondence holds only if the operator is linear , completely continuous , self-adjoint and it acts on Hilbert spaces . (= Extremely strict conditions must be imposed to both operators and spaces!) 10.3.3 Definition of self-adjoint operator Let us introduce an adjoint operator of a given linear operator T . The adjoint operator is, so to say, an operator version of an adjoint matrix with finite dimension. In elementary linear algebra, the adjoint matrix of an m -by-n complex matrix A is the n -by-m matrix A * obtained from A by taking the transpose and then taking the complex conjugate of each entry. For example, if A = 1 -2 -i 1 + i i , (10.30) then we have A * = 1 1 -i -2 + i -i . (10.31) So what kind of operators does adjoint matrices correspond to? Since no matrix-entry is assigned to operators, transposition-based definition is of no use for our aim. Instead, we pay attention to the following property of general adjoint matrices. ( Au , v ) = ( u , A * v ) (10.32) We make a concept-expansion on the basis of (10.32), which is the relation between two inner products. The need of inner product for the definition of adjoint matrices implies the need of inner product for defining their operator versions, too. This is why we consider operators in Hilbert spaces as candidates for operators having adjoint properties. Below is an exact definition of adjoint operators. Definition (Adjoint operator): Let T be a linear operator in a Hilbert space. The adjoint operator T * of T is defined by ( T ( u ) , v ) = ( u , T * ( v )) (10.33) 14 In order to realize the situation, we need to choose such f ∈ L 2 ([ 0 , 1 ]) that yields f ( x ) = 0 almost everywhere and f ( x ) , 0 only at the vicinity of x = α . Under this condition, T f and α f are effectively the same operators, similarly to the relation between Ae i and a i e i commented above.
- Michael A. Parker(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Therefore, Hermitian Operators produce complete sets of orthonormal vectors that can be taken as a basis set for the Hilbert space. The development follows that in T.D. Lee ’ s book listed in the chapter references. De fi nition 3.1: Bounded Operator Let ^ H be a Hermitian Operator in a Hilbert space with a complete orthonormal set (basis) given by B ¼fj f i i : i ¼ 1, 2, ::: g The operator ^ H is bounded from below if there exists a constant C (note that it will be a real number for a Hermitian Operator) such that for all vectors j f i in the Hilbert space h f j ^ H j f i h f j f i > C ( 3 : 79 a) The vector j f i in this case is not necessarily normalized. However, note that the vector j c ¼j f i = k f k is normalized to one and that h c j ^ H j c i¼ h f j ^ H j f i k f k 2 ¼ h f j ^ H j f i h f j f i > C ( 3 : 79 b) indicates that one can focus on vectors normalized to one (rather than the full vector space) for the bounded property. In effect, one looks to see how the operator affects vectors terminating on the ‘‘ unit sphere. ’’ 172 Solid State and Quantum Theory for Optoelectronics Example 3.49 Suppose ^ H is bounded from below, show ^ H is bounded from above. S OLUTION h f j ^ H j f i h f j f i > C ) h f j ^ H j f i h f j f i > C ) h f j ^ H j f i h f j f i < C Example 3.50 Suppose ^ H is Hermitian with eigenvectors fj n i for n ¼ 0, 1, . . . g and ^ H j n i¼ E n j n i and E 0 E 1 E 2 . Show E 0 must be the lower bound. Assume for this example that the eigenvectors form a basis although we shall show this for some special cases later in this section. S OLUTION In view of Equation 3.79b, consider only those vectors normalized to one. Then consider an arbitrary vector j c i (normalized to one) that can be expanded in the eigenvectors since we assume that they form a basis.
- Albert T. Jr. Fromhold(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
§8] EXPECTATION VALUES FOR QUANTUM-MECHANICAL OPERATORS 77 common definition, that J o p is Hermitian if (5°y)*/* = f*(£°*f) dx (1.232) for arbitrary /, by choosing / to be g t + yg h where y is an arbitrary complex parameter. It is more clear from the latter definition (1.232) that there is a correlation between physical observables and the Hermitian property of operators. That is, whenever/ s is an eigenfunction of J op corresponding to a real eigenvalue λ (so that λ* = λ & ), as must be the case when J op represents a physical observable, then the relation (1.232) reduces to [ (W/. A = [ /.WJ * (1-233) J Ω, J Ω, which is obviously satisfied. EXERCISE Prove that the definition (1.231) for a Hermitian Operator follows from the seemingly more restricted definition (1.232) of such. (Hint : Consider/to be a linear combination of g t and g } with complex coefficients.) To return to our proof, using the relation (1.231), Eq. (1.230) can be written I (^gd*gjd[ = gj gfgj dr. (1.234) Taking the complex conjugate of this relation gives (3 op g i )gJdr = q* Giu* dr. (1.235) The order of the scalar factors in the products in the integrand is not important. As mentioned before, q^ is real. This equation can therefore be written gf&»gidr = qj J Ω, g*9i dr. Subtracting from Eq. (1.229) gives immediately 0 = (qi - qj) gfgi dr. (1.236) (1.237) If gfi and gj correspond to different eigenvalues of ^ op , namely, q t Φ q j9 then this relation tells us that lj gi dr = 0 (i #7), (1.238) which proves the theorem and thereby justifies Eq. (1.219). If g t is a different function from g } but q t = q } , which is the degenerate case, Eq. (1.237) then gives no information on the value of Qx gjg t dr. This integral can still be zero if the functions g t and gj are chosen properly. Relating to this question, there is a
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