At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral) The image shows a mathematical expression for a potential energy function \( V(x) \). The function is defined piecewise as follows:\[ V(x) = \begin{cases} 0, & 0 < x < L \\\infty, & \text{elsewhere}\end{cases}\]Explanation:- The function \( V(x) \) represents a potential barrier.- It is equal to 0 within the interval \( 0 < x < L \).- Outside of this interval, the potential function \( V(x) \) is infinite (\(\infty\)), indicating an impenetrable barrier. This type of potential is often used in quantum mechanics to describe a particle in an infinite potential well, where the particle is free to move within the interval \( 0 < x < L \) but cannot exist outside of it. The function is defined as follows:\[\Psi(x) = \begin{cases} \left( \frac{1+i}{2} \right) \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) + \frac{1}{\sqrt{2L}} \sin\left(\frac{2\pi x}{L}\right), & 0 < x < L \\0, & \text{elsewhere}\end{cases}\]This describes a piecewise function, where \(\Psi(x)\) takes a specific form involving trigonometric functions and complex numbers for \(0 < x < L\), and is zero elsewhere.

Answered: Image /qna-images/answer/fdeebe55-1ed6-494b-a3c8-0dfa74a83c30.jpg

At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find for the particle at time t. (Hint: can be obtained by inspection, without an integral)

At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find for the particle at time t. (Hint: can be obtained by inspection, without an integral)

Similar questions

Consider the wavefunction for a particle in a one-dimensional box when the level is n = 6. Calculate the total probability of finding the particle between x = 0 and x = L/12? Provide your answer to three significant figures.
An electron has a kinetic energy of 13.3 eV. The electron is incident upon a rectangular barrier of height 21.5 eV and width 1.00 nm. If the electron absorbed all the energy of a photon of green light (with wavelength 546 nm) at the instant it reached the barrier, by what factor would the electron's probability of tunneling through the barrier increase?
For a quantum particle in a scattering state as it interacts a certain potential, the general expressions for the transmission and reflection coefficients are given by T = Jtrans Jinc R = | Jref Jinc (1) where Jinc, Jref, Jtrans are probability currents corresponding to the incident, reflected, and transmitted plane waves, respectively. (a). potential For the particle incident from the left to the symmetric finite square well -Vo; a < x < a, V(x) = 0 ; elsewhere, show that B Ꭲ ; R = A A
Calculate the uncertainties dr = V(x2) and op = V(p²) for %3D a particle confined in the region -a a, r<-a. %3D
Notice for the finite square-well potential that the wave function Ψ is not zero outside the well despite the fact that E < V0. Is it possible classically for a particle to be in a region where E < V0? Explain this result
a question of quantum mechanics: Consider a particle in a two-dimensional potential as shown in the picture Suppose the particle is in the ground state. If we measure the position of the particle, what isthe probability of detecting it in region 0<=x,y<=L/2 ?
A particle is initially prepared in the state of = [1 = 2, m = −1 >|, a) What's the expectation values if we measured (each on the initial state), ,, and Ĺ_ > b) What's the expectation values of ,, if the state was Î_ instead?
= = An electron having total energy E 4.60 eV approaches a rectangular energy barrier with U■5.10 eV and L-950 pm as shown in the figure below. Classically, the electron cannot pass through the barrier because E < U. Quantum-mechanically, however, the probability of tunneling is not zero. Energy E U 0 i (a) Calculate this probability, which is the transmission coefficient. (Use 9.11 x 10-31 kg for the mass of an electron, 1.055 x 10-34] s for h, and note that there are 1.60 x 10-19 J per eV.) (b) To what value would the width L of the potential barrier have to be increased for the chance of an incident 4.60-eV electron tunneling through the barrier to be one in one million? nm
Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)
A particle is described by the wavefunction Ψ(t, x), and the momentum operator is denoted by pˆ. a) Write down an expression for the differential operator pˆ. b) Write down an expression for the expectation value of the momentum, ⟨p⟩. c) Write down an expression for the probability density, ρ. d) Write down an expression for the probability of finding the particle between x = a and x = b.
3