Exam 40: One-Dimensional Quantum Mechanics

The lowest energy level of a particle confined to a one-dimensional region of space (a box,or infinite well)with fixed length L is E₀.If an identical particle is confined to a similar region with fixed length L/6,what is the energy of the lowest energy level that the particles have in common? Express your answer in terms of E₀.

A particle is confined to a one-dimensional box (an infinite well)on the x-axis between x = 0 and x = L.The potential height of the walls of the box is infinite.The normalized wave function of the particle,which is in the ground state,is given by ψ(x)= sin ,with 0 ≤ x ≤ L.What is the probability of finding the particle between x = 0 and x = L/3?

An electron is trapped in an infinite square well (a box)of width Find the wavelength of photons emitted when the electron drops from the n = 5 state to the n = 1 state in this system.(c = 3.00 × 10⁸ m/s,h = 6.626 × 10^-34 J ∙ s,m_el = 9.11 × 10^-31 kg)

An electron is in an infinite square well that is 2.6 nm wide.What is the smallest value of the state quantum number n for which the energy level exceeds 100 eV? (h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg,1 eV = 1.60 × 10^-19)

The smallest kinetic energy that an electron in a box (an infinite well)can have is zero.

An electron is in an infinite square well (a box)that is 2.0 nm wide.The electron makes a transition from the to the state,what is the wavelength of the emitted photon? (h = 6.626 × 10^-34 J ∙ s,m_el = 9.11 × 10^-31 kg,1 eV = 1.60 × 10^-19)

An electron is confined in a one-dimensional box (an infinite well).Two adjacent allowed energies of the electron are 1.068 × 10^-18 J and 1.352 × 10^-18 J.What is the length of the box? (h = 6.626 × 10^-34 J ∙ s,m_el = 9.11 × 10^-31 kg)

The wave function of an electron in a rigid box (infinite well)is shown in the figure.If the electron energy 98.0 eV,what is the energy of the electron's ground state? (m_el = 9.11 × 10^-31 kg)

An electron in an infinite potential well (a box)makes a transition from the n = 3 level to the ground state and in so doing emits a photon of wavelength 20.9 nm.(c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34J ∙ s,m_el = 9.11 × 10^-31 kg) (a)What is the width of this well? (b)What wavelength photon would be required to excite the electron from its original level to the next higher one?

A 3.10-eV electron is incident on a 0.40-nm barrier that is 5.67 eV high.What is the probability that this electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, m_el = 9.11 × 10^-31 kg,h = 1.055 × 10^-34 J ∙ s,h = 6.626 × 10^-34 J ∙ s)

Find the wavelength of the photon emitted during the transition from the second EXCITED state to the ground state in a harmonic oscillator with a classical frequency of 3.72 × 10¹³ Hz. (c = 3.00 × 10⁸ m/s,1 eV = 1.60 × 10^-19 J,h = 1.055 × 10^-34 J ∙ s,h = 6.626 × 10^-34 J ∙ s)

A lithium atom,mass 1.17 × 10^-26 kg,vibrates with simple harmonic motion in a crystal lattice,where the effective force constant of the forces on the atom is k = 49.0 N/m. (c = 3.00 × 10⁸ m/s,h = 6.626 × 10^-34 J ∙ s,h = 1.055 × 10^-34 J ∙ s,1 eV = 1.60 × 10^-19 J) (a)What is the ground state energy of this system,in eV? (b)What is the wavelength of the photon that could excite this system from the ground state to the first excited state?

An electron is in the ground state of an infinite well (a box)where its energy is 5.00 eV.In the next higher level,what would its energy be? (1 eV = 1.60 × 10^-19 J)

An electron is bound in an infinite square-well potential (a box)on the x-axis.The width of the well is L and the well extends from x = 0.00 nm to In its present state,the normalized wave function of the electron is given by: ψ(x)= sin (2πx/L).What is the energy of the electron in this state? (h = 6.626 × 10^-34 J ∙ s,m_el = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

An electron is bound in an infinite well (a box)of width 0.10 nm.If the electron is initially in the n = 8 state and falls to the n = 7 state,find the wavelength of the emitted photon. (c = 3.00 × 10⁸ m/s,h = 6.626 × 10^-34 J ∙ s,m_el = 9.11 × 10^-31 kg)

A particle confined in a rigid one-dimensional box (an infinite well)of length 17.0 fm has an energy level and an adjacent energy level E_n+1 = 37.5 MeV.What is the value of the ground state energy? (1 eV = 1.60 × 10^-19 J)

You want to confine an electron in a box (an infinite well)so that its ground state energy is 5.0 × 10^-18 J.What should be the length of the box? (h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg)

The atoms in a nickel crystal vibrate as harmonic oscillators with an angular frequency of 2.3 × 10¹³ rad/s.The mass of a nickel atom is 9.75 × 10^-26 kg.What is the difference in energy between adjacent vibrational energy levels of nickel? (h = 6.626 × 10^-34 J ∙ s, H = 1.055 × 10^-34 J ∙ s,1 eV = 1.60 × 10^-19 J)

A particle is confined to a one-dimensional box (an infinite well)on the x-axis between x = 0 and x = L.The potential height of the walls of the box is infinite.The normalized wave function of the particle,which is in the ground state,is given by ψ(x)= sin ,with 0 ≤ x ≤ L.What is the maximum probability per unit length of finding the particle?

The lowest energy level of a certain quantum harmonic oscillator is 5.00 eV.What is the energy of the next higher level?