Kindly answer no iii, iv & v 1. The general solution for a particle in a one-dimensional infinite potentialwell of width L is given asy(x) = C sin kx + Dcos kxwhere C, D and k are constants.(i)What are the boundary conditions at the walls of the well?Show that the coefficient k in the above solutions is(ii)NTk =Ln = 1, 2,3, -..(ii)Show that the wave function of the particle in the potential wellisпахw„(x) = C sin|Ln = 1,2,3, -..(iv) By normalisation, determine the value of C.(v)Prove that the energy of the particle inside the potential well isquantised.[Hint: sin? 0 = (1+sin 20).]

Answered: Image /qna-images/answer/5d57705f-b107-4792-b5a8-46f823808ec0.jpg

(iii) Show that the wave function of the particle in the potential well is NA X w.(x) = C sin n = 1,2,3, -.. %3D L (iv) By normalisation, determine the value of C. (v) Prove that the energy of the particle inside the potential well is quantised.

(iii) Show that the wave function of the particle in the potential well is NA X w.(x) = C sin n = 1,2,3, -.. %3D L (iv) By normalisation, determine the value of C. (v) Prove that the energy of the particle inside the potential well is quantised.

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

See similar textbooks

Related questions

Q: For the hydrogen atom, show that V100 and Y300 are orthogonal.

A: The detail explanation is given below

Q: What configurations must undergo the distortions predicted by the Jahn-Teller theorem? Why is…

Q: Calculate the expectation values of px and px2 for a particle in the state with n = 2 in a…

A: Where, H is the Planck’s constant The wave function for n = 2 in a square well potential is given by…

Q: Evaluate the average potential energy, , for the ground state (u = 0) of the harmonic oscillator by…

A: The wave function which can be used for determining the expression for potential energy is shown…

Q: Vanadium tetrachloride (VCl4 ) is a bright red colored liquid with a vapor pressure of 5x10-2 torr…

A: The orbital angular momentum of electron inn a given orbital is calculated by using the following…

Q: Assume that the states of the π electrons of a conjugated molecule can be approximated by the…

A: SOLUTION: Step 1: For a particle in one dimensional Box, the normalized Wave function is

Q: Calculate the energy separations in J, kJ/mol, eV (electron volts), and cm-1between the levels…

A: a) n=2 and n=1 of an electron in abox of length 1.50nm. For energy separations, we can use the given…

Q: Construct the potential energy operator of a particle with potential energy V(x)=1/2kfx2, where kf…

A: The information about the location of a particle is given by Born interpretation of the wave…

Q: (c) Estimate the wavelength of electrons that have been accelerated from rest through a potential…

Q: Which of the following functions can be normalized (in all cases the range for x is from x = −∞ to…

A: Given: Range for x is from -infinity to +infinity. (i) sin(ax) Substitute x= +infinity and…

Q: What is the maximum degeneracy of the wavefunctions of a particle confined to the interior of an…

A: A particle present in the interior of an icosahedral nanoparticle is same as the particle surrounded…

Q: Calculate the rotational energy of CO at J=2 given a bond length of 1.0 Å. unit in eV.

Q: The moment of inertia of an SF6 molecule is 3.07 × 10−45 kg m2. What is the minimum energy needed to…

A: Given: The moment of interia of an SF6 molecule is 3.07×10-45 kg m2.

Q: Calculate the expectation values of x and x2 for a particle in the state with n = 1 in a…

A: The expectation value of x2 for a particle is given by following expression. The value of ψ for one…

Q: (a) Show that all reciprocal lattice vectors of the form G= hÃ+ kB+IC are perpendicular to the…

Q: What is the degeneracy of a molecule rotating with J = 3?

A: The phenomenon by which a single energy level corresponds to two or more than two states of motion…

Q: can the function x2e-ax^2 be normalized in the range of -inf to inf, and if so what is the…

A: A wavefunction is normalized if:

Q: Consider a particle in the following 1D box. The potential in this box has the form: 1 2. V = kx² 2…

Q: Calculate the temperature at which there is 1% probabailty that a state with an energy 0.5eV, above…

A: The expression to find the Fermi energy is as follows: fE=1eε-μκT+1

Q: Normalize (to 1) the wavefunction e-ax^2 in the range −∞ ≤ x ≤ ∞, with a > 0. Refer to the…

A: Given: The wavefunction is e-ax^2 The range is −∞ ≤ x ≤ ∞ and a > 0.

Q: 1. The general solution for a particle in a one-dimensional infinite potential well of width L is…

Q: Suppose that you designed a spacecraft to work by photon pressure. The sai l was a completely…

Q: The speed of a certain proton is 0.45 Mm s−1. If the uncertainty in its momentum is to be reduced to…

A: Given Information - speed of proton = 0.45 Mm s-1 uncertainity in momentum = 0.0100 % = 0.0001 To…

Q: (a) If Â = 3x? and B = , then show that Â and ß donot commute with respect to the dx function f(x) =…

A: Commutator of A & B is generally represented as : The operation is possible with taking…

Q: Chemical potential in two dimensions. Show that the chemical potential of a Fermi gas in two…

Q: Q1: A state is described in terms of three vectors )|P) and ) with state: Iw)-t)-굼아)1이) of operator…

Q: A hydrogen atom. treated as a point mass. is confined to a one-dimensional square well of width 1.0…

Q: The particle on a ring is a useful model for the motion of electrons around the porphine ring, the…

Q: Evaluate the linear momentum and kinetic energy of a free proton described by the wavefunction e−ikx…

A: Wavefunction = e-ikx Where, k = 5 nm-1

Q: In the language of Problem 12C.5, show that if G(t) = (a cos ω1t + b cos ω2t)e−t/τ, then the…

A: The spectrum consists of two lines with intensities proportional to a and b located at ω = ω1 and…

Q: 4. The variance of many measurements of observable A is given by o = (A²) – (A)². Derive the…

A: We have to derive the measurement in the variation of the position of a particle in a box in a state…

Q: Calculate the wavlength (in nm) of electromagnetic radiation with a frequency of 5.8 x 1014 s-1.

A: Wavelength of an electromagnetic wave can be defined as the distance between one crest or trough to…

Q: Calculate the size of the quantum involved in the excitation of an electronic motion of frequency…

Q: 1. Calculate the wavelength and frequency at which the intensity of the radiation is a maximum for a…

A: The wavelength (λmax) of the maximum amount of radiation emitted by a black body at a certain…

Q: Calculate the separation between the two lowest energy levels for an N2 molecule in a one-…

A: Given, mass of N2, m = 28 amu = 28x1.6605x10-27 kg Length = 2 cm =…

Q: Confirm that the wavefunction Y(x,t)= e(px-Et)/h is a solution of the one-dimensional time- 2.…

Q: Given the following probability distribution for a discrete random variable X; X = x 1 2 3 4 P(X =…

A: A)It is known that the sum of probabilities of a probability distribution of random variables is…

Q: 3.2 Consider an assembly of six indistinguishable particles obeying Bose-Einstein statistics. The…

A: The solution of the given question is

Q: Find the uncertainty ox in x as a function of the uncertainties ou and oy in u and v for the…

A: Solution. Given that (a) x = 1/2(u + v) (b) x = uv^2 (c) x = u^2+v^2

Q: Estimate the wave l ength of electrons that have been accelerat ed from rest through a potential…

A: To estimate the wave length of electrons that have been accelerated from rest through a potential…

Q: Which of these choices is the expectation value of momentum p for a particle in the state n = 5 in a…

A: The reasonable expected value of the result of an experiment is known as its expected value.

Q: 4. The variance of many measurements of observable A is given by o? = (A²) – (A)². Derive the…

A: We have to derive the measurement in the variation of the position of a particle in a box in a state…

Concept explainers

Techniques and applications of quantum theory

Quantum theory, often known as quantum physics, is the foundation of modern physics and encompasses the nature and behavior of matter and energy at the atomic and molecular levels. As the exact nature and behavior of atomic and subatomic particles are impossible to determine, the laws of classical physics do not apply to this discipline. As a result, quantum field theory is primarily based on assumptions and probabilities.

Question

Kindly answer no iii, iv & v

1. The general solution for a particle in a one-dimensional infinite potential
well of width L is given as
y(x) = C sin kx + Dcos kx
where C, D and k are constants.
(i)
What are the boundary conditions at the walls of the well?
Show that the coefficient k in the above solutions is
(ii)
NT
k =
L
n = 1, 2,3, -..
(ii)
Show that the wave function of the particle in the potential well
is
пах
w„(x) = C sin|
L
n = 1,2,3, -..
(iv) By normalisation, determine the value of C.
(v)
Prove that the energy of the particle inside the potential well is
quantised.
[Hint: sin? 0 = (1+sin 20).]

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Techniques and Applications of Quantum Theory

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, chemistry and related others by exploring similar questions and additional content below.

Similar questions

(b) A system with chemical potential u = 12.5 meV contains identical particles each with integer spin and has a single quantum state at energy E, = 0 meV. The temperature of the system is fixed at 290 K. To three significant figures, what is the ratio Pg/P1, where P is the probability of finding a single particle in the state Eo, and P3 is the probability of finding 3 particles in this state? P3/P (Input your answer as a number.)
A 1.00 kg weight is traveling at a constant velocity of 10.0 ms-1, in a cubic room 3.00 m on an edge.(a). Determine the sum of the squares of its three quantum numbers.(b). What would be the ground state energy for this weight?
A two-level system is in a quantum state = α₁₁ + a22, which can be represented by the vector a = {a1, a2}. We are looking for the conditions under which is eigenstate of the operator c., defined below. 1) In the expression c.ỗ, c is a vector with components {Cr, Cy, Cz} ( C; are real numbers) and ỗ is a vector with components {σx, σy,σz} (σ¿ are 2 × 2 matrices). This means that the operator c. can also be represented by a 2 × 2 matrix. Write the matrix of the operator c. knowing that 0 0x = = (₁ }) = ( 5 ) . Oy σz= 0 (6-99) 1 0 1 0 2) In matrix form, the eigenvalue equation is AX AX, where A is the matrix of the operator of interest, X the column matrix representing the eigenvector and the corresponding eigenvalue. Write the eigenvalue equa- tion that needs to be verified for the quantum state to be an eigenvector of the operator c.. = 3) Note that AX = XX ⇒ (A - I)X 0, where I is the identity matrix. (AAI)X=0 is true if and only if det(A - I) = 0. Solve this equation for the operator…
Identify the systems for which it is essential to include a factor of 1/N! on going from Q to q : (i) a sample of carbon dioxide gas, (ii) a sample of graphite, (iii) a sample of diamond, (iv) ice.
4. The variance of many measurements of observable A is given by o = (A²) – (A)². Derive the variance in the measurements of position of a particle in a box in state „(x).
4. Measurement momentum yield (A) +1 ħ. (C) 0 ħ. of the z component of angular for an electron in the 2px state will (B) (D) -1 ħ. ±1 h.
2. A one-particle, one-dimensional system has the state function y=(sinat) ()* e/² + (cosat) (32/C) 1/4 2 1/4 e-x²/² е πρ >-x2/c2 xe where a is a constant and c = 2.000 Å. If the particle's position is measured at t = 0, estimate the probability that the result will lie between 2.000 Å and 2.001 Å.
Which of the following functions can be normalized (in all cases the range for x is from x= (-infinity) to (infinity), and a is a positive constant): (i) sin(ax); (ii) cos(ax)e-x^2? Which of these functions are acceptable as wavefunctions? Source: Atkins Physical Chemistry 11th Edition, E7B.4(b)
(a) Consider the translational motion of a particle in a one-dimensional box of mass m, that is free to move between x = 0 and x = L: (i) State the boundary conditions that must be satisfied. (1) (ii) Draw rough sketches of three different boxes to illustrate the relationship between the energy separation and the increasing or changing length of the box. (3)
E7B.3(b) Which of the following functions can be normalized (in all cases the range for x is from x = -∞ to ∞, and a is a positive constant): (i) sin(ax); (ii) cos(ax) ex²? Which of these functions are acceptable as wavefunctions?
A one-particle, one-dimensional system has the state function Y = (sinat) (2)+(cosat) (32) 14 2-x² / c² xe where a is a constant and c = 2.000 Å. If the particle's position is measured at t = 0, estimate the probability that the result will lie between 2.000 Å and 2.001 Å.
Calculate the relative populations of two quantum statesseparated by an energy of 0.4 × 10-21 J if the temperatureis 25°C.