The normalised wavefunction for an electron in an infinite 1D potential well of length 89 pm can be written: ψ=(-0.696 ψ2)+(0.245 i ψ9)+(g ψ4). If the state is measured, there are three possible results (i.e. it is in the n=2, 9 or 4 state). What is the probability (in %) that it is in the n=4 state?

The normalised wavefunction for an electron in an infinite 1D potential well of length 89 pm can be written: ψ=(-0.696 ψ2)+(0.245 i ψ9)+(g ψ4). If the state is measured, there are three possible results (i.e. it is in the n=2, 9 or 4 state). What is the probability (in %) that it is in the n=4 state?

Physics for Scientists and Engineers with Modern Physics

10th Edition

ISBN:9781337553292

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter40: Quantum Mechanics

Section: Chapter Questions

Problem 18P

See similar textbooks

Similar questions

Consider a particle confined to a 1-dimensional box of length L = 6 nm. A. What is the probability of locating the particle between x = 3 nm and x = 3.2 nm in the ground state? B. Evaluate T where is the normalized particle in a box wave function. If the particle is an electron and the quantum number of the state it's in (n) is 3, compute the eigenvalue you get with the formula you obtain. (J) C. What is the potential energy operator equal to for PIAB? 00 O Don't know... ○ 10
An electron is trapped in a one-dimensional infinite potential well that is 200 pm wide; the electron is in its ground state. What is the probability that you can detect the electron in an interval of width öx = 5.0 pm centered at x = 100 pm? (Hint: The interval öx is so narrow that you can take the probability density to be constant within it.) Number i Units
An electron is trapped in a one-dimensional infinite potential well that is 430 pm wide; the electron is in its ground state. What is the probability that you can detect the electron in an interval of width &x = 5.0 pm centered at x = 260 pm? (Hint: The interval Sx is so narrow that you can take the probability density to be constant within it.) Number i Units
Problem 3. Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation h² d²v 2m dx² EV, with boundary conditions (0) = 0 and (1) = 0. Second, the Quantum Harmonic Oscillator (QHO) V = EV h² d² 2m da² +ka²) 1 +kx² 2 (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system. (d) Explain why you can observe a particle outside of the "classically allowed region". Hint: you can use any state and compute an integral to determine a probability of a particle being in a given region.
Consider a particle trapped in a 1D box with zero potential energy with walls at x = o and x = L. The general wavefunction solutions for this problem with quantum number, n, are: V,6) = sin ) 4n(x) = The corresponding energy (level) for each wavefunction solution is: n²h? En 8mL? a) What is the probability of finding the particle between x = L/4 and x = 3L/4 when the particle is in quantum state n = 1, 2 and 3. You can use calculator or a numerical program to do the integral. For people who want to try doing the integral by hand, the following identity will be helpful: sin²(x) = (1 – cos (2x))/2.
A particle of mass 1.60 x 10-28 kg is confined to a one-dimensional box of length 1.90 x 10-10 m. For n = 1, answer the following. (a) What is the wavelength (in m) of the wave function for the particle? m (b) What is its ground-state energy (in eV)? eV (c) What If? Suppose there is a second box. What would be the length L (in m) for this box if the energy for a particle in the n = 5 state of this box is the same as the ground-state energy found for the first box in part (b)? m (d) What would be the wavelength (in m) of the wave function for the particle in that case? m
1.3 The effective potential of a particle is given by: Veff(r) = 2mr? + V(r). Find the effective force corresponding to this effective potential. 1.4 A hydrogen atom has an electron radial eigenfunction given by: u0,1(r) : (ezan, Obtain the probability density of this eigenfunction. 10
PROBLEM 2. Calculate the probabilities of measurement of different mo- menta p for a particle with the wave function (x) = Ceka sin (gx), %3D where C is a normalization constant.
We are going to use Heisenberg's uncertainty principle to estimate the ground- state energy of hydrogen. In our model, the electron is confined in a one- dimensional well with a length about the size of hydrogen, so that Ax = 0.0529 nm. Estimate Ap, and then assume that the ground-state energy is roughly Ap2/2me. (Give your answer in Joules or electron-volts.)
A harmonic oscillator of mass m and angular frequency w is in the initial state of wavefunction Y(x, 0) = Ai¢o(x) + 2Ai¢2(x) 3. a. Obtain the constant A b. Write the function Þ(x, t) c. Calculate the uncertainties Ax and Ap in the state of wavefunction (x, t) and show that the Heisenberg uncertainty principle is satisfied
A particle of mass m is contained in a one-dimensional infinite well extending from L to x=- 2 L The particle is in its ground state given by ø.(x)= /2/L cos(zx/L). X = - The walls of the box are moved suddenly to form a box extending from x = -L to x= L. what is the probability that the particle will be in the ground state after this sudden expansion?
10QM. Please answer question throughly and detailed.