Lecture physics a2 schrödinger’s equation and the particle in a box huynh quang linh

Lecture 7 Schrödinger’s Equation and the Particle in a Box U= y(x) 0 L U= n=1 n=2 x n=3 Content  Particle in a “Box” matter waves in an infinite square well  Wavefunction normalization  General p[.]

Lecture 7: Schrödinger’s

Equation and the Particle in a Box

U= 

y (x)

U= 

n=1

n=2

x

n=3

 Particle in a “Box” matter waves in an infinite square well

 Wavefunction normalization

 General properties of bound-state wavefunctions

 Notice that if U(x) = constant, this equation has the simple form:

) x (

C dx

d

2

2

y

 y

For positive C, what is the form of the solution?

For negative C, what is the form of the solution?

where is a constant that might be positive or negative.C 2m2 (UE)



a) sin kx b) cos kx c) eax d) e-ax

a) sin kx b) cos kx c) eax d) e-ax

KE term

PE term

Total E term

) ( )

( ) (

) (

2 2

x E

x x

U dx

x d

 

Last lecture: The time-independent SEQ (in 1D)

 Notice that if U(x) = constant, this equation has the simple form:

) x (

C dx

d

2

2

y

 y

For positive C, what is the form of the solution?

For negative C, what is the form of the solution?

where is a constant that might be positive or negative.C 2m2 (UE)



a) sin kx b) cos kx c) eax d) e-ax

a) sin kx b) cos kx c) eax d) e-ax

KE term

PE term

Total E term

) ( )

( ) (

) (

2 2

x E

x x

U dx

x d

 

Last lecture: The time-independent SEQ (in 1D)

Constraints on the form of y (x)

 y (x) 2 corresponds to a physically meaningful quantity –

the probability of finding the particle near x.

Therefore, in a region of finite potential:

y (x) must be finite, continuous and single-valued.

(because probability must be well defined everywhere)

d y /dx must be finite, continuous and single valued.

There is usually no significance to the sign of y (x).

(it goes away when we take the absolute square)

{In fact, we will see that y (x) can even be complex!}

Exercise 1

y( x )

x

(c)

y( x )

x

x

(b)

1 Which of the following hypothetical wavefunctions for a particle in some realistic potential U(x) is acceptable?

2 Which of the following wavefunctions corresponds to a particle more likely to be found on the left side?

y(x)

y(x)

(c)

y(x)

(b) Acceptable

Both y(x) and dy/dx

are continuous everywhere

(a) Not acceptable

y(x) is not

continuous at x=0.

dx

d y not defined.

dy/dx is not continuous at x=0

(c) Not acceptable

y(x)

x

(c)

y(x)

x

x (b)

1 Which of the following hypothetical wavefunctions for a particle in some realistic potential U(x) is acceptable?

2 Which of the following wavefunctions corresponds to a particle more likely to be found on the left side?

y(x)

y(x)

(c)

y(x)

None of them!

(a) is clearly completely symmetric.

(b) might seem to be “higher” on

the left than on the right, but it

is only the absolute square the

determines the probability

y 2

This is a basic problem in “Nano-science” It’s a simplified (1D) model for an electron confined in a quantum structure (e.g., “quantum dot”), which scientists/engineers make, e.g., at the UIUC Microelectronics Laboratory !

Application of SEQ: “Particle in a Box”

KE term

PE term

Total E term

U = 0 for 0 < x < L

U =  everywhere else

) ( )

( ) (

) (

2 2

x E

x x

U dx

x d

 

(www.kfa-juelich.de/isi/) (newt.phys.unsw.edu.au)

‘Quantum dots’

U(x)

 As a specific important example, consider a quantum particle confined

to a small region, 0 < x < L, by infinite potential walls We call this a

“one-dimensional (1D) box”

 Recall, from last lecture, the time-independent SEQ in one dimension:

Waves: Boundary conditions

 Boundary condition: Constraints on a wave where the potential changes

 E = 0 at surface of a metal film Displacement = 0 for wave on string

E = 0

 If both ends are constrained (e.g., for a cavity of length L), then only certain wavelengths l are possible:

n l = 2L

n = 1, 2, 3 …

‘mode index’

3 2L/3

4 L/2

L

Rope Demo