Particle in a Box and Tunneling. A Particle in a Box Boundary Conditions The Schrodinger Equation Tunneling Through a Potential Barrier Homework

Size: px

Start display at page:

Download "Particle in a Box and Tunneling. A Particle in a Box Boundary Conditions The Schrodinger Equation Tunneling Through a Potential Barrier Homework"

Harry Hancock
6 years ago
Views:

1 Particle in a Box and Tunneling A Particle in a Box Boundary Conditions The Schrodinger Equation Tunneling Through a Potential Barrier Homework

2 A Particle in a Box Consider a particle of mass m and velocity v confined by two impenetrable walls. Classically, the particle bounces back and forth along the x- axis with constant momentum and kinetic energy, and there are no restrictions on the values of the particle s energy and momentum. A quantum mechanical description of the particle s motion requires that we find the appropriate wave function consistent with the conditions of the problem. Figure (a)

3 Boundary Conditions for the Particle in a Box Because the walls are impenetrable, the probability of finding the particle outside the box is zero, so we must have Since the potential is zero inside the box, we start with the wave function for a free particle! "$#%## Figure (b)

4 Wave Function for the Particle in a Box & ')(+*, -./10 & ')(+*, -./10 & '4(+*, -./ , 8:9 ;<9=>9@?@?@? Figure 28.21

5 Energy of the Particle in a Box A, B, D, "E F D,, B 32, B C GIH $J, B 2 H $J B K H J H 7 7, 8:9 ;<9=>9@?@?@? Figure 28.22

6 Boundary Conditions and Quantization We saw that for the particle in the box, the application of the boundary conditions L quantized the energy of the system. It turns out that any constraint on the motion of particles in a system represents one or more boundary conditions that results in quantization of the energy of the system.

7 ; E The Schrodinger Equation The wave functions that describe a quantum system are the solutions of a wave equation developed by Erwin Schrodinger in The time-independent version of the Schrodinger equation for a particle of mass J moving along the x-axis under the influence of a potential energy function M is P & N O &, D & PQ( R S Note that the Schrodinger equation states that the total energy is the sum of the kinetic energy and the potential energy and is conserved: T + M = U = constant.

8 & \ Requirements for Allowable Solutions must be continuous must be single-valued and V W2 V must be continuous for finite values of M must approach zero as x approaches + or - infinity, so that obeys the normalization XZY condition PQ(, 8 [ Y \

9 ^ Particle in a Box from the Schrodinger Equation ] ] In the region, where M, we can write the Schrodinger equation as [ a ^ a b where ^dcfe g V_^ V [ $J U ` ^ The general solution to this equation has the form h i a X j kml a Applying the boundary condition at, we have i + X j kml n X j Which yields the particular solution o a Applying the boundary condition at, we have o a Which gives us the quantization condition a p $J U ` q! "$#%#$#

10 ^ z Particle in a Box from the Schrodinger Equation (cont d) Solving for the quantized energy, we have Usr B ^ K J ^ q! "$#%#$# The allowed wave functions are r t u Applying the normalization condition vxw y w z^ V we get the normalization constant 2{ The normalized wave functions are then h i t u

11 Example A particle of mass m is confined to a one-dimensional box between and. Find the expectation value of the position of the particle for a state with quantum number.

12 Example A particle of mass m is confined to a one-dimensional box between and. Find the expectation value of the position of the particle for a state with quantum number. (~}, X Y [ Y & ( & PQ(, (./ PQ( (~}, X Y [ Y (./ PQ(,

Tunneling Through a Potential Barrier Consider a particle with energy U that encounters a square potential barrier of height M and width. Classically, the particle would be reflected by the barrier.

13 Tunneling Through a Potential Barrier Consider a particle with energy U that encounters a square potential barrier of height M and width. Classically, the particle would be reflected by the barrier. Quantum mechanically, there is a finite probability that the particle tunnels through the barrier. The probability that the particle tunnels through the barrier is given by the transmission coefficient, while the probability that it is reflected is given by the reflection coefficient ƒ. Figure 28.23

14 ˆ, ; E ' S Tunneling (continued) Since the incident particle is either transmitted or reflected, we must have +ƒ =1. When the barrier is very wide or very high (M > > U ), < < 1 and an approximate expression for can be obtained [ q where O N D *

15 Homework Set 9 - Due Mon. Apr. 19 Read Sections Answer Question Do Problems 28.35, 28.36, 28.37, & 28.45

Opinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability

Opinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic