(a)
Plot of wave function as a function of
(a)
Answer to Problem 62CP
The wave function was plotted as a function of
Explanation of Solution
Write the given wave function of the electron.
Here,
Write the formula to calculate the probability of finding a particle in a certain range.
Here,
Refer equation (I) and plot the wave function as a function of
Figure 1 below shows the plot of wave function.
Conclusion:
The wave function was plotted as a function of
(b)
Plot of probability density as a function of
(b)
Answer to Problem 62CP
The probability density was plotted as a function of
Explanation of Solution
Write the given wave function of the electron.
Here,
Write the formula to calculate the probability of finding a particle in a certain range.
Here,
Refer equation (II) and plot the probability density as a function of
Figure 2 below shows the plot of wave function.
Conclusion:
The probability density was plotted as a function of
(c)
To show that
(c)
Answer to Problem 62CP
The
Explanation of Solution
Write the given wave function of the electron.
Here,
For the wave function to be a reasonable wave function, there are set of condition.
The
As
The
The
Conclusion:
The
(d)
To normalize the wave function.
(d)
Answer to Problem 62CP
The normalization constant of the given wave function is
Explanation of Solution
Write the given wave function of the electron.
Here,
Write the condition for normalized wave function.
The wave function is symmetric. Thus re-write the above condition.
Substitute equation (I) in the above equation.
Conclusion:
The normalization constant of the given wave function is
(e)
The probability of finding the electron in the range
(e)
Answer to Problem 62CP
The probability of finding the particle in the range
Explanation of Solution
Refer section (d) and write the given normalized wave function of the electron.
Here,
Write the formula to calculate the probability of finding a particle in a certain range.
Here,
Refer equation (II) in equation (III) to determine probability in range
Conclusion:
The probability of finding the particle in the range
Want to see more full solutions like this?
Chapter 41 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
- For a "particle in a box" of length, L, the wavelength for the nth level is given by An 2L %3D 2п and the wave function is n(x) = A sin (x) = A sin (x). The energy levels are пп %3D n?h? given by En : %3D 8mL2 lPn(x)|2 is the probability of finding the particle at position x in the box. Since the particle must be somewhere in the box, the integral of this function over the length of the box must be equal to 1. This is the normalization condition and ensuring that this is the case is called “normalizing" the wave function. Find the value of A the amplitude of the wave function, that normalizes it. Write the normalized wave function for the nth state of the particle in a box.arrow_forwardA hypothetical one dimensional quantum particle has a normalised wave function given by (x) = ax - iß, where a and 3 are real constants and i = √-1. What is the most likely. x-position, II(x), for the particle to be found at? 0 11(x) == ○ II(2) = 0 ○ II(r) = 2/ O II(z) = 011(r) = ± √ +√ 13 aarrow_forwardDetermine ψ∗ψ for the following wave functions:a) ψ(θ) = sin θ + icosθ b) ψ(x) = eiaxarrow_forward
- What is the value N so that ψ(x) = N/(a2 + x2) can give rise to a valid probability density?arrow_forwardThe probability density function (PDF) for electrons to be detected on the x-axis between 0 nm and 1.0 nm is shown below. What is the probability of finding the electron between x = 0.5 nm and x = 1.0 nm? |w(x)* (nm') 2.0 1.0 0.5 x (nm) 1.0arrow_forwardA particle is moving in a three-dimensional potential V (x, y, z)=-mo (2x² +y² +4z° ). If the mass of the particle is m, then what is the energy of the particle in the lowest quantum State? (a) (3+ v2 (b) ħo (c) (3+ V2) ħo (d) ħo 12arrow_forward
- A quantum particle is described by the wave function ψ(x) = A cos (2πx/L) for −L/4 ≤ x ≤ L/4 and ψ(x) everywhere else. Determine:arrow_forwardA quantum system is described by a wave function (r) being a superposition of two states with different energies E1 and E2: (x) = c191(r)e iEit/h+ c292(x)e¯iE2t/h. where ci = 2icz and the real functions p1(x) and p2(r) have the following properties: vile)dz = ile)dz = 1, "0 = rp(x)T#(x)l& p1(x)92(x)dx% D0. Calculate: 1. Probabilities of measurement of energies E1 and E2 2. Expectation valuc of cnergy (E)arrow_forwardAn electron moving in a box of length ‘a’. If Z1 is the wave function at x1 = a/4 with n=1 and Z2 at x = a/4 for n=2 find Z1/Z2arrow_forward
- Let Ψ (x, t) = (A / (a2 + x2)) exp (-i 2 π E t / h ) be a normalized solution to Schrodinger’s equationfor constants A, a, and E.(a) What is A in terms of a?(b) What is the potential function V(x)?(c) Evaluate Δx Δp. Is the uncertainty principle satisfied?arrow_forwardIf an electron is in the n=3 state l =1, list the possible quantum states (n, l, ml,ms ).arrow_forwardAn electron in an infinitely deep square well has a wave function that is given by ψ3(x) = √2/Lsin (3πx/L)for 0 ≤ x ≤ L and is zero otherwise. (a) What are the most probable positions of the electron? (b) Explain how you identify them.arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningModern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
- Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning