The initial wave function of a free particle is:
Ψ(x,0) = A, for |x|
= 0, otherwise
where a and A are positive real numbers. The particle is in a zero (or constant) potential environment since it is a free particle
a) Determine A from normalization.
b) Determine φ(p) = Φ(p,0), the time-zero momentum representation of the particle state. What is Φ(p,t)? Sketch φ(p). Locate the global maximum and the zeros of φ(p). Give the expression for the zeros (i.e., for the location of the zeros).
c) Determine the momentum space probability density |Φ(p, t)|2 and show then that Φ(p, t) is normalized in momentum space. (You can use a table integral.) Sketch |Φ(p,t)|2 and locate the global maximum and the zeros. Give the expression for the zeros.
d) Crudely estimate σx and σp for time zero. Are the results consistent with the Heisenberg uncertainty principle?
e) Is it possible to find the group velocity? If so, what is it?
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0,...
A. Momentum space We define the momentum space wave function φ(p) as where Ψ(x)is a solution of the Schrödinger equation in configuration (position) space a) Show that the expectation values of and p can be written in terms of Ф(p) as <p(p)p(p)dp b) Demonstrate that φ(p) is normalized, ie if Ψ(x) is normalized. J ΙΨ(2)12dr-1 c) Show that Ф(p) 2dp can be interpreted as the probability to find a particle with momen tum between p and p+ dp
Consider a particle confined to one dimension and positive z with the wave function 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: f) Calculate the expectation value of the momentum, (p) via the canonical expression -0o g) Calculate the expectation value of (p) via the canonical expression h) Use your results for(i) and (pay to calculate the variance in...
1: For a free particle, U(x)-0, in the state Ψ(x)-e®, determine if the particle is in an eigenstate of the following operators and give the eigenvalue if it is. a position z m b. momentum c. Energy d. Translation (D)
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0) = 1 /√5 [2 ?₁ (?) + ?₂ (?)] where ?1 and ?2 are the eigenfunctions of the oscillator Hamiltonian for ? = 1,2 states. a) Write down the expression for Ψ(?,?). b) Calculate the probability density ℙ(?,?) = |Ψ(?,?)| ² . Express it as a sinusoidal function of time. To simplify the result, let ? ≡ (?² ℏ)/ 2??² . c) Calculate 〈?〉...
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression: x <0,x>2L (A) Determine the value of the normalization constant c. (B) Draw the wavefunction. (C) Calculate Prob(L/2 S x 3 3L/2), the probability of finding the particle between x - L/2 and 3L/2 Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression:...
Consider a particle confined to one dimension and positive r with the wave function 0, z<0 where N is a real normalization constant and o is a real positive constant with units of (length)-1. For the following, express your answers in terms of a: a) Calculate the momentum space wave function. b) Verify that the momentum space wave function is normalized such that (2.4) c) Use the momentum space wave function to calculate the expectation value (p) via (2.5)
A particle is represented by the following wave function: ψ(x) =0 x<−1/2 ψ(x) =C(2x + 1) −1/2 < x < 0 ψ(x) =C(−2x + 1) 0 < x < +1/2 ψ(x) =0 x > +1/2 (a)Evaluate the probability to find the particle between x=0.19 and x=0.35. (b) Find the average values of x and x2, and the uncertainty of x: Δx=√(x2)av-(xav)2 xav= (x2)av= Δx =
5. A free particle has the initial wave function, where A and a are positive real constants. (a) Normalize ψ(x,0). (b) Find φ(k). (c) Construct $(z,t), in the forn of an integral. (d) Discuss the limiting cases (a very large, and a very small).
Problem 3: A free particle of mass m in one dimension is in the state Hbr Ψ(z, t = 0) = Ae-ar with A, a and b real positive constants. a) Calculate A by normalizing v. b) Calculate the expectation values of position and momentum of the particle at t 0 c) Calculate the uncertainties ΔΧ and Δ1) for the position and momentum at t 0, Do they satisfy the Heisenberg relation? d) Find the wavefunction Ψ(x, t) at a...
Problem 2: Time development of a free particle The wave function of a free particle at time t- 0 is given by exp(2K1T Now answer the questions below. l. what is the time evolved wave function ψ(z,t) ? points 2. What is the average momentum at any future time? 4 points 3. What is the average energy at any future time ? 3 points Problem 2: Time development of a free particle The wave function of a free particle at...