Consider a particule in the box trial wavefunction φ(x)=(2/L)^1/2(sin(πx/L)+αsin(2πx/L)). Set up the expression for the energy functional Ε and optimize it with respect to the coefficient α.
Consider a particule in the box trial wavefunction φ(x)=(2/L)^1/2(sin(πx/L)+αsin(2πx/L)). Set up the expression for the energy...
Consider a trial wavefunction for an electron in H atom in the following form º(r) = re-ar where a is an adjustable parameter. Optimize a so that you obtain the minimum energy (i.e., find the extremum by imposing .. (E) = 0). How does the minimum energy compare to the ground state energy of an electron? Hint: n! IX"e-ax dx = for α> 0 Δ η Ε Ν an+1 Integration of function f(r,0,0) in spherical coordinates: - po T 27...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
√x-2 set up Consider carve ye ² Sin (2 VX-2). parameterize this curre by letting X = 2+t² but dont evaluate an integeral Wates with respect to t. to compute the length of the portion of the curve from points x = 24 to X = 13
In class, we considered a box with walls at \(x=0\) and \(x=L\). Now consider a box with width \(L\) but centered at \(x=0\), so that it extends from \(x=-L / 2\) to \(x=L / 2\) as shown in the figure. Note that this box is symmetric about \(x=0 .\) (a) Consider possible wave functions of the form \(\psi(x)=A \sin k x\). Apply the boundary conditions at the wall to obtain the allowed energy levels.(b) Another set of possible wave functions...
If you were to use a trial function of the form φ(x):: (1 + cax 2)r"r/2, where α (kμ/h2)1/2 and c is a variational parameter, to calculate the ground-state energy of a harmonic oscillator, what do you think the value ofc will come out to be? Why? 8-6.
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
Consider a perturbed particle in a box, with potential energy: for x <-L/2 2brx/L for -1/2sxSL2 for x >L/2 nd confining the zero order functions to n-1,2, 3, 4 (i.e. the lowest four Using Matrix algebra, and confining the zero order functions to n solutions to the unperturbed particle in a box problem) determine the energy of the l (Hint: In diagonalizing a matrix, you may reorder the quantum numbers in any way you like). d) Consider a perturbed particle...
The expression Φ(x, h)-(1-2xh + h2)-1/2 where |hl < 1 is the generating function for Legendre polynomials. φ(x, h) can be expressed as a sum of Legendre polynomials The function (x, h) = Po(x) + hA(x) + h2Pg(x) + hn (x) The generating function of the Legendre polynomials has some applications in Physics, such as expressing the electric potential at point P due to a charge q. The location of the charge is r with respect to the origin O...