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1. Show that -rlao Rio (r) =-ne-ri, 3/2 0 Is a solution of R: 0 and that it is normalized. 1. Show that -rlao Rio...
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The sys...
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The system has a set of orthonormal energy eigenfunctions (), 2(x),.. with corresponding eigenvalues E, E2, .... Write down the overlap rule for the probability of getting the energy E when the energy is measured at time t 0 (b) Suppose that a system is described by a normalized wave function of the form (,0) an(r), where the an are...
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0)...
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if N ...
3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if N is a submodule of M then N is cyclic of order s for some s \Ir Conversely, if s | r show that M has a cyclic submodule of order s.
3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if...
Show that a rigid rotor for m = 0 and l = 0 is normalized.
Show that a rigid rotor for m = 0 and l = 0 is normalized.
Bonus: Maximum Power transfer i R RI. a. Show that R -Ri for maximum power transfer...
Bonus: Maximum Power transfer i R RI. a. Show that R -Ri for maximum power transfer in the circuit above (Use Calculus) b. What is the relationship between Zh and Zu in the circuit below? ac circuit
2. a. Show that y² + x – 3 = 0 is an implicit solution to...
2. a. Show that y² + x – 3 = 0 is an implicit solution to dy/dx = -1/(2y) on the interval (-0, 3). b. Show that xy3 – xy: sin x = 1 is an implicit solution to dy_(x cos x + sin x - 1) y 3(x - x sin x) on the interval (0, /2).
ri 0 2-t] 3. Let Az = 0 t 1 .v= 10 0 2 (1) Find...
ri 0 2-t] 3. Let Az = 0 t 1 .v= 10 0 2 (1) Find all possible t such that A, has determinant 1. (7 po (2) Find all possible t such that v is in the row space of Aų. (3) Find all possible t such that v is in an eigenspace of Al.
2. (1 pt) Assume we observe pairs (X1, Yİ), . . . , (X,X,), Let (Ri,...
2. (1 pt) Assume we observe pairs (X1, Yİ), . . . , (X,X,), Let (Ri, S), . . . , (R,S,) be their corresponding ranks. The definition of Kendall's τ is ncnd n(n-1), where ne and nd are the number of concordant pair and discordant pairs (see equation (3.7) of the lecture note). Show that this definition is equivalent to where
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any sm...
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any small ε > 0.]
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε)...
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP...
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The system has a set of orthonormal energy eigenfunctions (), 2(x),.. with corresponding eigenvalues E, E2, .... Write down the overlap rule for the probability of getting the energy E when the energy is measured at time t 0 (b) Suppose that a system is described by a normalized wave function of the form (,0) an(r), where the an are...
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if N is a submodule of M then N is cyclic of order s for some s \Ir Conversely, if s | r show that M has a cyclic submodule of order s.
3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if...
Bonus: Maximum Power transfer i R RI. a. Show that R -Ri for maximum power transfer in the circuit above (Use Calculus) b. What is the relationship between Zh and Zu in the circuit below? ac circuit
2. a. Show that y² + x – 3 = 0 is an implicit solution to dy/dx = -1/(2y) on the interval (-0, 3). b. Show that xy3 – xy: sin x = 1 is an implicit solution to dy_(x cos x + sin x - 1) y 3(x - x sin x) on the interval (0, /2).
ri 0 2-t] 3. Let Az = 0 t 1 .v= 10 0 2 (1) Find all possible t such that A, has determinant 1. (7 po (2) Find all possible t such that v is in the row space of Aų. (3) Find all possible t such that v is in an eigenspace of Al.
2. (1 pt) Assume we observe pairs (X1, Yİ), . . . , (X,X,), Let (Ri, S), . . . , (R,S,) be their corresponding ranks. The definition of Kendall's τ is ncnd n(n-1), where ne and nd are the number of concordant pair and discordant pairs (see equation (3.7) of the lecture note). Show that this definition is equivalent to where
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any small ε > 0.]
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε)...
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
4.22. Consider the vibrating system described by 42 -2 1 Compute the mass-normalized stiffness matrix, the eigenvalues, the normalized eigen- vectors, the matrix P, and show that PTMP I and PTKP is the diaggaal matrix of eigenvalues Л
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