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

A Particle in a Box� Consider a particle of mass m and velocity v confined by twoimpenetrable walls.� Classically, the particle bounces back and forth along the x-axis with constant momentum and kinetic energy, and thereare no restrictions on the values of the particle’s energy andmomentum.� A quantum mechanical description of the particle’s motion re-quires that we find the appropriate wave function consistentwith the conditions of the problem.

Figure 28.20 (a)

Boundary Conditions for the Particle in a Box� Because the walls are impenetrable, the probability of findingthe 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 28.20 (b)

Wave Function for the Particle in a Box& ')(+* , - .�/10 ��� ��& ')(+* , - .�/10 ��� �� �32��& '4(+* , - .�/10 5 � � �� 6 7 , 8:9�;<9�=>9@?@?@?

Figure 28.21

Energy of the Particle in a BoxA , B� , B� �32�� , � B� CD � , ��"E F � , GIH�$J , � � B 2 � �� H�$JD � , B HK J � H 7 � 7 , 8:9�;<9�=>9@?@?@?

Figure 28.22

Boundary Conditions and Quantization� We saw that for the particle in the box, the application of theboundary conditions

� ����� � � �L�� � �quantized the energy

of the system.� It turns out that any constraint on the motion of particles ina system represents one or more boundary conditions that re-sults in quantization of the energy of the system.

The Schrodinger Equation� The wave functions that describe a quantum system are thesolutions of a wave equation developed by Erwin Schrodingerin 1926.� The time-independent version of the Schrodinger equation for

a particle of mass J moving along the x-axis under the influ-

ence of a potential energy function M � ���is

N O �; EP � &PQ( � R S & , D &

� Note that the Schrodinger equation states that the total energyis the sum of the kinetic energy and the potential energy andis conserved: T + M = U = constant.

Requirements for Allowable Solutions� � � ��� must be continuous� � � ��� must be single-valued and V �W2 V � must be continuous forfinite values of M � ���

� � � ��� must approach zero as x approaches + or - infinity, so

that� � ���

obeys the normalization conditionXZY[ Y \ & \ � PQ( , 8

Particle in a Box from the Schrodinger Equation� In the region� ] � ] �

, where M � �, we can write the

Schrodinger equation asV_^ �V � ^ � [ �$J U` ^ � � [ a ^ �where a � b ^dcfeg� 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 �$J U` � � � � � � �q� � �! "�$#%#$#

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 conditionvxw

y w z � z ^ V � � �we get the normalization constant

� � � 2{�� The normalized wave functions are then� � �h�i� �� ����� t � � �� u

Example

A particle of mass m is confined to a one-dimensional box be-tween

� � �and

� � �. Find the expectation value of the position�

of the particle for a state with quantum number�

.

Example

A particle of mass m is confined to a one-dimensional box be-tween

� � �and

� � �. Find the expectation value of the position�

of the particle for a state with quantum number�

.

| (~} , X Y[ Y & ��(�& PQ( , �� ( �� .�/�0 5 � � �� 6 � PQ(| (~} , �� X Y[ Y ( .�/10 � 5 � � �� 6 PQ( , � �

Tunneling Through a Potential Barrier� Consider a particle with energy U that encounters a squarepotential 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 isgiven by the transmission coefficient � , while the probabilitythat it is reflected is given by the reflection coefficient � .

Figure 28.23

Tunneling (continued)� Since the incident particle is either transmitted or reflected, wemust have � + � =1.� When the barrier is very wide or very high ( M > > U ), � < < 1

and an approximate expression for � can be obtained� � � [ �q� �where � , ; E ' S N D *

O

Homework Set 9 - Due Mon. Apr. 19� Read Sections 28.10 - 28.13� Answer Question 28.16� Do Problems 28.35, 28.36, 28.37, 28.44 & 28.45