1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show...
3. Show that the differential equation 1 d de sin sin 0 de de sin20 Lastl leads to the associated Legendre equation if we consider the c= cos0, A- v(v+1), ux)-e(0)
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
please show calculations Solve the equation on the interval 0 s < 2t. 1) 2 cos 0+32 2) tan2 = 3 3) 2 sin2 = sino show calculation please 4) 2 cos2 - 3 cos 0+1=0 5) sin2 - Cos2 0 = 0 Simplify the expression 6) + tan e 1+ sin e cose 7) (1 + cot e)(1-cote) -sce Establish the identity. 8) (sin x)(tan x cos x - cotx cos x) = 1 - 2 cos2x 9) (1...
Please show working and explanation if possible. Thank you so much in advance! The first four spherical harmonics Ym are given below. r-e'a cose 3 11/2 Y. 3 11/2 3 1/2 sin θ e_ίφ a) Give the mathematical functions that describe the s orbital and the pz orbital. (10 marks) b) Unsöld's theorem states that the spherical harmonics satisfy the relation +I IYİm12 = constant m--l t Unsöld's theorem is valid for the family of spherical harmonics with (15 marks)...
From Arfken, demostrate equation 12.85. Step by step solution please. Associated Legendre Polynomials The regular solutions, relabeled pn (x), are (12.73c) These are the associated Legendre functions.16 Since the highest power of x in Pn (x) is xn, we must have m n (or the m-fold differentiation will drive our function to zero) In quantum mechanics the requirement that m n has the physical interpretation that the expectation value of the square of the z component of the angular momentum...
6.16 The deuteron is a nucleus of heavy hydrogen consisting of ne proton and one neutron.As a simple model for this nucleos consider a single particle of mass m moving in a fixed spehreical -symmetric potencial V(r) defined by V(r)=-Vo for rro .This is called a spherical square well potential.Asssume that the particle is in a bound state with l=0 a) Fin the general solutions R(r) to the radical Schoringer euatio for rro.Use the fact that the wave function must...
1) Amass, m, on a spring with spring constant k obeys the equation of motion Where-1 kg. Andk is assigned a value 1 (in Sl units) What are the units of the spring constant? Assuming that at time O, the mass mis at rO traveling with a velocity of 1 m/s Work out the maximum displacement of the mass in subsequent oscillations Can you find an alternative way of getting this answer? 2) Amass,, on a spring with spring constant...
Problem 3. Show that the solution of the partial differential equation (Laplace equation), Wxx(x, y) + Wyy(x, y) = 0, with the four boundary conditions: w(x,0) = 0, w(x, 1) = 0, w(0, y) = 0 and w(1, y) = 24 sin ny, can be obtained as w(x, y) = 2 sinh nx · sin ny. [Suggested Solution Steps for Problem 3] (1) Apply the method of separation of variables as w(x,y) = X(x) · Y(y); (2) substitute into the...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w:= cos e given by 2 F" (r) +2rF (r) - n(n + 1) F, (r) = 0, (1 w2)G (w) - 2wG", (w) +n(n +1)G, (w) 0. (n 0,1,2,....
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 where 0 is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w =cos e given by 12 F" (T) +2r F (r) - n(n + 1 ) Fr (r) = 0, (1 w2)G (w) - 2wG (w) + n(n+ 1)G, (w)...