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4. Let T be the time reversal operator. Show that T=U*K where U is the unitary...
Let V be R, with thestandard inner product. If is a unitary operator on V, show...
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
I need help with 1.b) 1a. Assume the unitary operator U-exp(1?/4lâ??? +â?â01. show that the matrix...
I need help with 1.b)
1a. Assume the unitary operator U-exp(1?/4lâ??? +â?â01. show that the matrix relation for the beam splitter, is identical with the unitary transformation Hint: use the Baker-Hausdorf lemma on page 13 of Gerry/Knight. 1b. operator U-exp(??/2la??, + âtaol). Obviously, for ?-?/2 the result in 1a is retrieved. Calculate the beam splitter matrix corresponding to the more general unitary
8) Let U be a unitary matrix. Show that: a. U is normal. b. Ifa is...
8) Let U be a unitary matrix. Show that: a. U is normal. b. Ifa is an eigenvalue of U then N= 1 (Hint: 1 Uxl =1 xl for every x e C”.
8) Let U be a unitary matrix. Show that: a. U is normal. b. If2 is...
8) Let U be a unitary matrix. Show that: a. U is normal. b. If2 is an eigenvalue of Uthen = 1 (Hint: Uxl =1 xl for every x € C".
8) Let U be a unitary matrix. Show that: a. U is normal. b. If 2...
8) Let U be a unitary matrix. Show that: a. U is normal. b. If 2 is an eigenvalue of Uthen 2=1 (Hint: 1 Uxl l xl for every xC".
Let U E Mat(n; C) be a unitary matrix. Show that (UzUy) = (y) for any...
Let U E Mat(n; C) be a unitary matrix. Show that (UzUy) = (y) for any 7,7 € C", where U acts by the usual matrix multiplication and (-:-) is the standard inner product on Ch.
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the gen...
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the generalized uncertainty principle for the operator and H σ//σ,2 |I What can you deduce about the value of in a stationary state? b) The time dependent Schroedinger equation allows the identification E-i Derive the uncertainty relation ACEAO). 2m
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the generalized uncertainty principle for the operator and H σ//σ,2 |I...
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall that...
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall that we have shown in class that in this case, there exists a unique unitary operator U such that A=UA. The point of this exercise is to prove the following result: an invertible operator A is normal if and only if U|A= |AU. a) Show that if UA = |A|U, then AA* = A*A. Now, we want to show the other direction, i.e. if AA*...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator...
Let V be an inner product space and u, w be fixed vectors in V . Show that T v = <v, u>w defines a linear operator in V . Show that T has an adjoint, and describe T ∗ explicitly
Let H be a complex Hilbert space. 6. (a) Let φ, ψ E H \ {0} . Define the linear operator T on H by Using the Cauchy...
Let H be a complex Hilbert space. 6. (a) Let φ, ψ E H \ {0} . Define the linear operator T on H by Using the Cauchy-Schwarz inequality, show that llll = Hell ll [4 marks] (b) A bounded linear operator A is said to have rank one if there exists v e H [0 such that for any u E H we have Au cu, where cu E C is a constant depending on u. (i) Show that...
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
I need help with 1.b)
1a. Assume the unitary operator U-exp(1?/4lâ??? +â?â01. show that the matrix relation for the beam splitter, is identical with the unitary transformation Hint: use the Baker-Hausdorf lemma on page 13 of Gerry/Knight. 1b. operator U-exp(??/2la??, + âtaol). Obviously, for ?-?/2 the result in 1a is retrieved. Calculate the beam splitter matrix corresponding to the more general unitary
8) Let U be a unitary matrix. Show that: a. U is normal. b. Ifa is an eigenvalue of U then N= 1 (Hint: 1 Uxl =1 xl for every x e C”.
8) Let U be a unitary matrix. Show that: a. U is normal. b. If2 is an eigenvalue of Uthen = 1 (Hint: Uxl =1 xl for every x € C".
8) Let U be a unitary matrix. Show that: a. U is normal. b. If 2 is an eigenvalue of Uthen 2=1 (Hint: 1 Uxl l xl for every xC".
Let U E Mat(n; C) be a unitary matrix. Show that (UzUy) = (y) for any 7,7 € C", where U acts by the usual matrix multiplication and (-:-) is the standard inner product on Ch.
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the generalized uncertainty principle for the operator and H σ//σ,2 |I What can you deduce about the value of in a stationary state? b) The time dependent Schroedinger equation allows the identification E-i Derive the uncertainty relation ACEAO). 2m
4) Problems on uncertainty relations. Notation: for any operator O let a-AO) a) Show that the generalized uncertainty principle for the operator and H σ//σ,2 |I...
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall that we have shown in class that in this case, there exists a unique unitary operator U such that A=UA. The point of this exercise is to prove the following result: an invertible operator A is normal if and only if U|A= |AU. a) Show that if UA = |A|U, then AA* = A*A. Now, we want to show the other direction, i.e. if AA*...
Let H be a complex Hilbert space. 6. (a) Let φ, ψ E H \ {0} . Define the linear operator T on H by Using the Cauchy-Schwarz inequality, show that llll = Hell ll [4 marks] (b) A bounded linear operator A is said to have rank one if there exists v e H [0 such that for any u E H we have Au cu, where cu E C is a constant depending on u. (i) Show that...
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