Q.M3 Home work 1 Due date 8.11.15 1
... 1) Is this state normalized? If yes,prove it. If not, normalize it. 2)Find a state |Bi that is orthogonal to |Ai. Make sure |Bi is normalized. 3) Express |hi and |si in the {|Ai, |Bi} basis. 4) What are possible outcomes of a hardness measurement on the state |Ai, and with what probability will each ...
... 1) Is this state normalized? If yes,prove it. If not, normalize it. 2)Find a state |Bi that is orthogonal to |Ai. Make sure |Bi is normalized. 3) Express |hi and |si in the {|Ai, |Bi} basis. 4) What are possible outcomes of a hardness measurement on the state |Ai, and with what probability will each ...
Free Fields, Harmonic Oscillators, and Identical Bosons
... and hence ↠â = n̂ and [â, ↠] = 1. Similarly, the direct product of single-mode Hilbert spaces in eq. (46) looks like a system of many harmonic oscillators, one oscillator for each mode β. This allows us to construct a whole family of oscillator-like creation and annihilation operators in the ...
... and hence ↠â = n̂ and [â, ↠] = 1. Similarly, the direct product of single-mode Hilbert spaces in eq. (46) looks like a system of many harmonic oscillators, one oscillator for each mode β. This allows us to construct a whole family of oscillator-like creation and annihilation operators in the ...
Waves and the Schroedinger Equation
... • For an operator Ô, with wavefunctions, ψn related as: Ô ψn = an ψn • The functions are known as eigenfunctions and the a n are eigenvalues. • The eigenvalues for quantum mechanical operators are real-valued since they correspond to experimental observables. ...
... • For an operator Ô, with wavefunctions, ψn related as: Ô ψn = an ψn • The functions are known as eigenfunctions and the a n are eigenvalues. • The eigenvalues for quantum mechanical operators are real-valued since they correspond to experimental observables. ...
generalized numerical ranges and quantum error correction
... S(H) such that span { I, A1 , A2 , A3 } has dimension 4, there is always an A4 ∈ S(H) for which Λ1 ( A1 , . . . , A4 ) is not convex. In the following, we show that Λk (A) is always star-shaped if dim H is sufficiently large. Moreover, it always contains the convex hull of Λk̂ (A) for k̂ = (m + 2)k. ...
... S(H) such that span { I, A1 , A2 , A3 } has dimension 4, there is always an A4 ∈ S(H) for which Λ1 ( A1 , . . . , A4 ) is not convex. In the following, we show that Λk (A) is always star-shaped if dim H is sufficiently large. Moreover, it always contains the convex hull of Λk̂ (A) for k̂ = (m + 2)k. ...
The Use of Fock Spaces in Quantum Mechanics
... Formal Definition of a Fock Space Definition A Fock space for bosons is the Hilbert space completion of the direct sum of the symmetric tensors in the tensor powers of a single-particle Hilbert space; while a Fock space for fermions uses anti-symmetric tensors. For the sake of simplicity, in this t ...
... Formal Definition of a Fock Space Definition A Fock space for bosons is the Hilbert space completion of the direct sum of the symmetric tensors in the tensor powers of a single-particle Hilbert space; while a Fock space for fermions uses anti-symmetric tensors. For the sake of simplicity, in this t ...
Quantum Field Theory on Curved Backgrounds. II
... the corresponding Euclidean quantum field theory makes use of Osterwalder-Schrader (OS) positivity [16, 17] and analytic continuation. On a curved background, there may be no proper definition of time-translation and no Hamiltonian; thus, the mathematical framework of Euclidean quantum field theory ...
... the corresponding Euclidean quantum field theory makes use of Osterwalder-Schrader (OS) positivity [16, 17] and analytic continuation. On a curved background, there may be no proper definition of time-translation and no Hamiltonian; thus, the mathematical framework of Euclidean quantum field theory ...
MATH3385/5385. Quantum Mechanics. Handout # 5: Eigenstates of
... Example 2: Finite one-dimensional well In example 1, we have seen a situation where the spectrum is discrete leading to bound states only, whereas in example 2 we have seen the other extreme situation, where the spectrum is fully continuous leading to scattering states characterised by the reflectio ...
... Example 2: Finite one-dimensional well In example 1, we have seen a situation where the spectrum is discrete leading to bound states only, whereas in example 2 we have seen the other extreme situation, where the spectrum is fully continuous leading to scattering states characterised by the reflectio ...
Lecture-XXIV Quantum Mechanics Expectation values and uncertainty
... It does not mean that if one measures the position of one particle over and over again, the average of the results will be given by On the contrary, the first measurement (whose outcome is indeterminate) will collapse the wave function to a spike at the value actually obtained, and the subsequent me ...
... It does not mean that if one measures the position of one particle over and over again, the average of the results will be given by On the contrary, the first measurement (whose outcome is indeterminate) will collapse the wave function to a spike at the value actually obtained, and the subsequent me ...
Symmetries and conservation laws in quantum me
... Using the action formulation of local field theory, we have seen that given any continuous symmetry, we can derive a local conservation law. This gives us classical expressions for the density of the conserved quantity, the current density for this, and (by integrating the density over all space) th ...
... Using the action formulation of local field theory, we have seen that given any continuous symmetry, we can derive a local conservation law. This gives us classical expressions for the density of the conserved quantity, the current density for this, and (by integrating the density over all space) th ...
Morse Theory is a part pf differential geometry, concerned with
... forms. We will also define the Hodge operator, which will be used in the proof of the Morse Inequalities. If we choose a Riemannian metric on the manifold M, denoted gm , this implies that for all m M , we have an inner product gm (, ) on the tangent space T ( M ). For smooth vector fields on ...
... forms. We will also define the Hodge operator, which will be used in the proof of the Morse Inequalities. If we choose a Riemannian metric on the manifold M, denoted gm , this implies that for all m M , we have an inner product gm (, ) on the tangent space T ( M ). For smooth vector fields on ...
Isometric and unitary phase operators: explaining the Villain transform
... Finally, we extend U to all of H by putting it zero on {ran|A|}⊥ = ker|A|, the orthogonal complement of ran|A|, and we are done. In passing we note that in our case the Hilbert space H is the finite-dimensional space s,ϕ as given by (11). Incidentally, there is an C|2S+1 = [|m, −S m S], isomor ...
... Finally, we extend U to all of H by putting it zero on {ran|A|}⊥ = ker|A|, the orthogonal complement of ran|A|, and we are done. In passing we note that in our case the Hilbert space H is the finite-dimensional space s,ϕ as given by (11). Incidentally, there is an C|2S+1 = [|m, −S m S], isomor ...
