we have
[ , ] = i
[ , ] = i
[ , ] = i
x = i
and 2 = 2 + 2 + 2
[ 2 , ] = [ , ] + [ , ]
= i [ + ]
finally
[ 2 , ] = 0
2 , are commute
the Hamiltonian equation is
= - (2/2m) ( 2/r2 + (2/r) (/r) ) + ( 2 / 2 m r2 ) + V ( r ) = 0
[ , i ] = 0
for a central potential it will be commute
[ , 2 ] = 0
so
, , 2 are commute
Given = î + ĵ − 2 and = î − 4 ĵ − , calculate the vector product ✕ . --i+---j+---k
(a) Show that (@) = sin e- is an eigenfunction of both Î, and Î", where = -1 1 a 1 22 sin + sin 020 sin0 and derive the corresponding eigenvalues. You may use the identity 1 a 1 sin sin 2 sin sin 0 80 sino 31 (sin 00 (5 marks) (6) Consider the function $(,0,4)= A - 1/200 sin 6e-ip, 20 where A is a constant and an is the Bohr radius. This is a hydrogen atom...
The Hamiltonian of a harmonic oscillator is: 2m 2 Show that: (1) [ł, î] =ihp () [f.h]=-ihmoʻx
2. Suppose we are given data on n observations (zi, y), î i, . . . , n, and we have a linear model, so that E (Y,) = Ao +Ari. Let A = SXY/Sxx and A,-F-Ax be the least-square estimates given in lecture. (a) Show that E(SXY)-ASxx and E(y)-Ao +AT. (b) Use (a) to show that E (A)-A and E(A)-A- In other words, these are unbiased estimators (c) The fitted values Yī = β0+812 i are used as estimates...
Given M = 4 i +6 - 2 K and N = 6 î - 49 - 4 , calculate the vector product M N. Need Help? Read It Watch It
Material: 8.3.2 Consider the matrix (1 2 3 A-2 3 1 (8.3.28) (i) Use (8.3.27) to find the dominant eigenvalue of A. (ii) Check to see that u-(1 , I , î ), is a positive eigenvector of A. Use 11 and Theorem 8.6 to find the dominant eigenvalue of A and confirm that this is exactly what was obtained in part 0) obtained in part (i) or(ii ii) Compute all the eigenvalues of A directly and confirm the result...
Vector field F = î 3y + ŷ (5 – 2x) + î (22 – 2) is given. Find: (e) The surface integral of the normal component of the curl of F over the open hemisphere x + y2 + z = 4 above the x-y plane.
Consider the following vectors. A = (1.70 m) î + (3.00 m) Î B: 3.90 m, at +65.0° C = (-4.00 m) i – (5.70 m) Î Õ: 5.00 m, at -230° (a) What is the sum of the four vectors in unit-vector notation? (b) What is the sum of the four vectors as a magnitude and an angle? magnitude direction • (counterclockwise from the positive x axis) m
The mean commute time for all commuting students of a university is 23 minutes with a population standard deviation of 4 minutes. A random sample of 63 driving times of commuters is taken. ̅ a) [2pts] Is the sampling distribution of the sample mean ? normal? Circle the number of i. ii. iii. iv. b) the best answer. Yes, because the sample size n is greater than 30. No, because the parent population of the data is not said to...
1 2 & 3 Q₂ = mbul Q₂:+12uL Q.: +30 Pam î om com 3 I 3M 6m 1. Find the magnitude and direction of the resultant electric field intensity at point A 2 Find the absolute potential at point A. 3. Determine the resultant force on charge R. Ra