1 Time-independent Schrödinger equation (TISE) Remember the (one-dimensional) time-independent Schrödinger equation (TISE) for a state (...
8. The time independent Schrödinger equation (TISE) in one-dimension where m is the mass of the particle, E ita energy, (z) the potential (a) Consider a particle moving in a constant pote E> Vo, show that the following wave function is a solution of the TISE and determine the relationahip betwoen E an zero inside the well, ie. V(2)a 0foros L, and is infinite ou , ie, V(x)-w (4) Assuming (b) Consider an infinite square well with walls at 1-0...
Potential energy function, V(x) = (1/2)mw2x2 Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation, then c ψ(x), where c is an arbitrary complex constant, is also a solution.
Instead of assuming that a one-dimensional particle has no energy (v(x)=0), consider the case of a one-dimensional particle which has finite, but constant, energy V(x)= V sub zero.. Show that the ID particle in a box wave functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger equation for this potential, and determine the energies En Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
(a) Write down the Schrödinger equation in Dirac notation. (2%) (b) Write x-representation in one dimensional systems. (3%) down both time dependent and time independent Schrödinger equations in
1. The time-dependent Schrödinger equation The time-dependent Schrödinger equation is -R2 824(1,t) + V (1,t) (1,t) = in 2m 0:2 . (a) For V1, t) = 0, show that the wave function (1,t) = A sin (kr - wt) does not satisfy the time- dependent Schrödinger equation. (b) For VI,t) = 0, Show that I, t) = A cos(kr - wt) + i sin (kr - wt) does satisfy this equation. This is a simple demonstration that the wavefunction in...
4. Consider the time-independent Schrödinger equation for an "atom" in which the attractive force between the electron and the proton is modeled as a spring. Then V(r)- (1/2)mu22, where m is the mass of the electron and w is the natural frequency of oscillation You're goal is to determine the eigenenergies of the electron and the corresponding wave functions, as outlined below Let's again start with the radial equation associated with the Schrodinge equation 4.37 in Griffiths [where u(r) R(r)...
Solve the following problems HW9. Show that the time-independent Schrödinger equation is given by P(x)/(x) = Eve from the traveling wave equation and the wave function (x./)=v(x)cos or HW10. Example 9.3 Calculation of a normalization factor Given that the wavefunction for the hydrogen atom in the ground state (n = 1) is of the form = Ne , where r is the distance from the nucleus to the electron and do is the Bohr radius, calculate the normalization factor N.
1) Consider a particle with mass m confined to a one-dimensional infinite square well of length L. a) Using the time-independent Schrödinger equation, write down the wavefunction for the particle inside the well. b) Using the values of the wavefunction at the boundaries of the well, find the allowed values of the wavevector k. c) What are the allowed energy states En for the particle in this well? d) Normalize the wavefunction
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...