Book edition 2nd Edition

Author(s) Randy Harris

Pages 633 pages

ISBN 9780805303087

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher (U₀ is increased) until essentially infinite.

Wavelength will reduce & energy of the particle in the ground state will increase when the walls of the well are made higher.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step-1: Penetration Depth

The penetration depth is the reciprocal of a factor $(α)$ , which represents how far the wave function, representing the particle inside the well, extends in the classically forbidden region outside the well. It is given as-

$δ = \frac{1}{α} = \frac{ℏ}{\sqrt{2 m (U_{o} - E)}} \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (1)$

Here, m and E are the mass and energy of the particle inside the well, role="math" localid="1660113324667" $U_{o}$ is the height of the well, and $ℏ$ is the modified Planck’s constant.

Step-2: Wave Function

For a particle in the ground state, equation (1) states that the penetration depth is inversely proportional to the square root of the difference in the height of the well and the energy of the particle. As the ground state energy of the particle remains constant, on increasing the walls of the well, wave function in the ground state will penetrate less and less into the classically forbidden region outside of the walls and when the well is infinitely high, the penetration depth is zero.

Step-3: Energies

On increasing the height of the well, the penetration depth decreases. This results in a decrease in the wavelength of the wave inside the well. Hence, the energy increases.

Thus, the penetration of wave function in the forbidden region decreases, and the energy of the wave increases, increasing the height of the well.

Find Study Materials for

Create Study Materials

Select your language

Modern Physics

Answers without the blur.

Short Answer

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher (U₀ is increased) until essentially infinite.

Step by Step Solution

Step-1: Penetration Depth

Step-2: Wave Function

Step-3: Energies

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Modern Physics

Answers without the blur.

Short Answer

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher (U0 is increased) until essentially infinite.

Step by Step Solution

Step-1: Penetration Depth

Step-2: Wave Function

Step-3: Energies

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher (U₀ is increased) until essentially infinite.