1) Wave function for the ground state of an harmonic oscillator is given by. (x) =...
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscillator, respectively. (a) Normalize (r, 0). (b) Find the wavefunction (r, t) at a later time t and hence evaluate (x, t) 2. Leave your answers involving expressions in to and ₁. c) sing the following normalized expression of...
The wave function of the ground state of a harmonic oscillator, with a force constant k and mass m is given as 1 Vo(x) = (1) where mwo k h m Calculate the probability of finding the particle outside the classical region. a = =
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2 The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
1. Variational method In this problem, you will approximate the ground state wave function of a quantum system using the variational theory. Use the trial wave function below 2 cos/T) , 1x1 trial a/2 to approximate the ground state of a harmonic oscillator given by 2.2 2 using a as an adjustable parameter. (a) Calculate the expectation value for the kinetic energy, (?) trial 4 points (b) Calculate the expectation value for the potential energy, Virial. Sketch ??tria, (V)trial, and...
For the ground state of a quantum harmonic oscillator, given by ?_0 = (?^(-1/2)*?^(-1/4))*?^(-(x^2)/(2?^2)) , show that the expectation values for potential and kinetic energy are equal.
Harmonic Oscillator: Determine the expectation value of the position of a harmonic oscillator in its ground state, Show that the uncertainty in the position of a ground state harmonic oscillator is Delta x 1/square root 2 (h^2/mk)^1/4.
Quantum Chemistry. Thx in Advance! 1. For a harmonic oscillator with unit mass and unit frequency, the Schrödinger equation for its eigenfunction is given by where n 0, 1, 2, . . .. Answer the following questions. Given a trial wave function, ?(x)-?000CnUn(x), where expression for the expectation value is is assumed to be real, the Prove that Eo2 h/2 2. Assume that the trial wave function for the ground state eigenfunction in Eq. (1) is ?(x) = cos Xx,...
2. The ground state wave function for the spherical harmonic oscillator with zero angular momentum is estimated to be (n) = e-ar Given that H = om deres + mw? mo? Calculate the optimal value of "a" that gives the lowest energy from dE/da. Please note that: and wtw) = L* de e-dor= VERA (.ly) = L º de c** L-am et + zma*r') e-*** (0917we) = Contact met ) VER