The solution of the Schrödinger equation obtained from the solution
... use of the Heisenberg equations, without noticing that a similar procedure is applicable to any Hamiltonian (not necessarily time-independent or quadratic in the coordinates and momenta) if one assumes that the solution of the Heisenberg equations is known, regardless of whether it has the same form ...
... use of the Heisenberg equations, without noticing that a similar procedure is applicable to any Hamiltonian (not necessarily time-independent or quadratic in the coordinates and momenta) if one assumes that the solution of the Heisenberg equations is known, regardless of whether it has the same form ...
Basic Quantum Mechanics in Coordinate, Momentum and
... between the coordinate, momentum and phase space representations of quantum mechanics. First, the ground‐state coordinate space eigenfunction for the harmonic oscillator is used for several traditional quantum mechanical calculations. Then the coordinate wave function is Fourier transformed into the ...
... between the coordinate, momentum and phase space representations of quantum mechanics. First, the ground‐state coordinate space eigenfunction for the harmonic oscillator is used for several traditional quantum mechanical calculations. Then the coordinate wave function is Fourier transformed into the ...
Coherent, Squeezed, and Thermal State of Harmonic Oscillator with
... harmonic oscillator in thermal state. Substitution of Eq. (63) into the above equation gives ...
... harmonic oscillator in thermal state. Substitution of Eq. (63) into the above equation gives ...
4. Introducing Conformal Field Theory
... A conformal field theory (CFT) is a field theory which is invariant under these transformations. This means that the physics of the theory looks the same at all length scales. Conformal field theories cares about angles, but not about distances. A transformation of the form (4.1) has a different int ...
... A conformal field theory (CFT) is a field theory which is invariant under these transformations. This means that the physics of the theory looks the same at all length scales. Conformal field theories cares about angles, but not about distances. A transformation of the form (4.1) has a different int ...
Minimal normal measurement models of quantum instruments
... measures (POVMs), that is, (weakly) σ-additive mappings M : Σ → L(H) such that M(X) ≥ 0 for all X ∈ Σ and M(Ω) = 1H , where 1H stands for the identity operator on H. If ΩN = {x1 , x2 , ...} with N ≤ ∞ elements and ΣN = 2ΩN is the corresponding σ-algebra of a POVM M, then M can be viewed as a collect ...
... measures (POVMs), that is, (weakly) σ-additive mappings M : Σ → L(H) such that M(X) ≥ 0 for all X ∈ Σ and M(Ω) = 1H , where 1H stands for the identity operator on H. If ΩN = {x1 , x2 , ...} with N ≤ ∞ elements and ΣN = 2ΩN is the corresponding σ-algebra of a POVM M, then M can be viewed as a collect ...
Geometry, Quantum integrability and Symmetric Functions
... A symmetric function P = (Pk )k≥0 is a sequence of symmetric polynomials such that for all k ≥ 1, Pk (y1 , . . . , yk−1 , 0) = Pk−1 (y1 , . . . , yk−1 ). Are there symmetric functions sλ (for every unbounded Young diagram λ) such that sI is the equivalence class of sλ (y1 , . . . , yk ) for all k, n ...
... A symmetric function P = (Pk )k≥0 is a sequence of symmetric polynomials such that for all k ≥ 1, Pk (y1 , . . . , yk−1 , 0) = Pk−1 (y1 , . . . , yk−1 ). Are there symmetric functions sλ (for every unbounded Young diagram λ) such that sI is the equivalence class of sλ (y1 , . . . , yk ) for all k, n ...
Introduction to random matrices
... defined by specifying a Hamiltonian on phase space) of fixed energy E and volume V. The motivation for this measure is that after specifying the energy of the system, every point in phase space lying on the energy surface should be equally likely since we have "no further macroscopic information." I ...
... defined by specifying a Hamiltonian on phase space) of fixed energy E and volume V. The motivation for this measure is that after specifying the energy of the system, every point in phase space lying on the energy surface should be equally likely since we have "no further macroscopic information." I ...
An introduction to rigorous formulations of quantum field theory
... field theories we use. Each approach presents coherent axioms but few working examples. The axioms might begin,“The mathematical data of a quantum field theory consists of a Hilbert space H, a self-adjoint Hamiltonian operator H acting on H,” and so on. Given a particular quantum field theory with s ...
... field theories we use. Each approach presents coherent axioms but few working examples. The axioms might begin,“The mathematical data of a quantum field theory consists of a Hilbert space H, a self-adjoint Hamiltonian operator H acting on H,” and so on. Given a particular quantum field theory with s ...
Quantum Energy–based P Systems - Computational Biology and
... The action of the operator G on Φ = c ...in |xi1 , . . . , xin i, expressed as a linear combination P of the elements of the n–register basis, is obtained by linearity: G(Φ) = ci1 ...in G(|xi1 , . . . , xin i). We recall that linear operators which act on n–registers can be represented as order 2n s ...
... The action of the operator G on Φ = c ...in |xi1 , . . . , xin i, expressed as a linear combination P of the elements of the n–register basis, is obtained by linearity: G(Φ) = ci1 ...in G(|xi1 , . . . , xin i). We recall that linear operators which act on n–registers can be represented as order 2n s ...
quantum computing for computer scientists
... vectors, which is critical for quantum computing. Ternary logic is different from traditional Boolean algebra (states of 0/1) because the latter has no third state, and hence confounds the states “the opposite of one” and “nothing”. Since addition is commutative (a + b = b + a) but subtraction is no ...
... vectors, which is critical for quantum computing. Ternary logic is different from traditional Boolean algebra (states of 0/1) because the latter has no third state, and hence confounds the states “the opposite of one” and “nothing”. Since addition is commutative (a + b = b + a) but subtraction is no ...
L. Snobl: Representations of Lie algebras, Casimir operators and
... The commutator (18) prevents the operators L̂j , K̂j from forming a Lie algebra. Nevertheless, this bothersome property can be circumvented if we consider a given energy level, i.e. a subspace HE of the Hilbert space H consisting of all eigenvectors of Ĥ with the given energy E. Operators L̂j , K̂j ...
... The commutator (18) prevents the operators L̂j , K̂j from forming a Lie algebra. Nevertheless, this bothersome property can be circumvented if we consider a given energy level, i.e. a subspace HE of the Hilbert space H consisting of all eigenvectors of Ĥ with the given energy E. Operators L̂j , K̂j ...
Weak Values in Quantum Measurement Theory
... To construct the general framework of the weak values advocated by Aharonov and his collaborators, which are experimentally accessible by the shift of the probe wave function in weak measurement. To show the efficiency of our proposed ...
... To construct the general framework of the weak values advocated by Aharonov and his collaborators, which are experimentally accessible by the shift of the probe wave function in weak measurement. To show the efficiency of our proposed ...
Chapter 9 Angular Momentum Quantum Mechanical Angular
... Notice like the nonsense operators hardness and color, none of the angular momentum component operators commute and none of the eigenvectors correspond. Also comparable, L2 is proportional to the identity operator, except in three dimensions. We can do something similar to the “hardness, color” case ...
... Notice like the nonsense operators hardness and color, none of the angular momentum component operators commute and none of the eigenvectors correspond. Also comparable, L2 is proportional to the identity operator, except in three dimensions. We can do something similar to the “hardness, color” case ...
QFT on curved spacetimes: axiomatic framework and applications
... of its representation on a specific Hilbert space as a unital C*-algebra. The observables are the selfadjoint elements, and the possible outcomes of measurements are elements of their spectrum. The spectrum spec(A) of A ∈ A is the set of all λ ∈ C such that A − λ1 has no inverse in A. One might susp ...
... of its representation on a specific Hilbert space as a unital C*-algebra. The observables are the selfadjoint elements, and the possible outcomes of measurements are elements of their spectrum. The spectrum spec(A) of A ∈ A is the set of all λ ∈ C such that A − λ1 has no inverse in A. One might susp ...
The Quantum Mechanics of Angular Momentum
... though it has angular momentum and, since it is charged, a magnetic moment led to the injection of the concept into the then still developing field of quantum mechanics. Wolfgang Pauli later named it 'spin' which is still used today. An interesting side story is that Ulenbeck and Goudsmit sent their ...
... though it has angular momentum and, since it is charged, a magnetic moment led to the injection of the concept into the then still developing field of quantum mechanics. Wolfgang Pauli later named it 'spin' which is still used today. An interesting side story is that Ulenbeck and Goudsmit sent their ...
Quantum error correcting codes and Weyl commutation relations
... for all ρ̂ of the form (1.1). Then the pair (C, R) is called a quantum N -correcting code. If a subspace C admits a recovery operation R so that (C, R) is a quantum N -correcting code we then say that C, or equivalently, the orthogonal projection P on C is a quantum N -correcting code. The dimension ...
... for all ρ̂ of the form (1.1). Then the pair (C, R) is called a quantum N -correcting code. If a subspace C admits a recovery operation R so that (C, R) is a quantum N -correcting code we then say that C, or equivalently, the orthogonal projection P on C is a quantum N -correcting code. The dimension ...
chapter 10. relation to quantum mechanics
... element 0 = Φ−1 (∅) and a greatest element 1 = Φ−1 (Y ). A logic is called a σ-logic if it is closed under countable applications of ∧ and ∨. The peculiarity of quantum systems is that their logics are non-distributive: e.g., the proposition “a and (b or c)” need not have the same truth value as “(a ...
... element 0 = Φ−1 (∅) and a greatest element 1 = Φ−1 (Y ). A logic is called a σ-logic if it is closed under countable applications of ∧ and ∨. The peculiarity of quantum systems is that their logics are non-distributive: e.g., the proposition “a and (b or c)” need not have the same truth value as “(a ...