The general form of the Schrödinger equation can be expressed as H EΨ = Ψ Consider the one-dimensional time dependent form:

2

2 ( , ) ( , ) ( , ) ( , )2

x t V x t x t i x tm x t

∂ ∂− Ψ + Ψ = Ψ

∂ ∂ ---- (1)

if V is a constant in t or V is a constant in x, then ( , ) ( ) ( )x t f x g tΨ = and equation (1) becomes

2

2

2

2

( ) ( ) ( , ) ( ) ( ) ( ) ( )2

1 1( ) ( , ) ( )2 ( ) ( )

f x g t V x t f x g t i f x g tm x t

f x V x t i g t Em f x x g t t

∂ ∂− + =

∂ ∂∂ ∂

− + =∂ ∂

=

where E is a constant. For the space-independent equation,

1 ( )( )

ln ( )

ln (0)

Eg t tg t i

Eg t t Ci

g C

∂ = ∂

= +

=

∫ ∫

thus, ( ) (0) (0)E iEt tig t g e g e

−= =

For the time-independent equation,

2

2

1 ( ) ( )2 ( )

f x V x Em f x x

∂− +

∂= ---- (2)

V(x) Particle in a box Consider the case where V(x) = 0 for 0 x L≤ ≤ equation (2) becomes

22 2

2( ) ( ) ( )mEf x f x k f xx∂

= − = −∂

0 L x

which is an 2nd order differential equation (ode) with the real general solution:

CKW2010 1

( ) sin cosf x A kx B k= + x

ubstituting,

S

(0) 0f B= = ( ) sin 0f L A kL

kL nnkL

ππ

= ==

here n is an integer

=

w

2 22

2 2

2 2 2

2

2

2n

n mkL

nEmL

π

π

= =

=

E

( ) sinnnf x xLπ⎛ ⎞= ⎜ ⎟

⎝ ⎠ and

thus,

( , ) ( ) ( ) sin (0)iE tnx t f x g t A x g e

Lπ −⎡ ⎤⎡ ⎤⎛ ⎞Ψ = = ⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦

Generally, for a particle trapped in a one dimensional “box”

( , ) ' siniE t

nnx t A eLπ− ⎛ ⎞Ψ = ⎜ ⎟

⎝ ⎠x ---- (3)

Since 1

2

0

| ( , ) |L

n x t dxΨ =∫ ,

[ ]

2 2

0

2

0

2

02

' sin 1

' 21 cos 12

' 2sin 12 2

' 12

2'

L

L

L

nA x dxL

A n x dxL

A L nx xn L

A L

AL

π

π

ππ

⎛ ⎞ =⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞− =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞− =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

=

=

∫

∫

and equation (3) becomes

CKW2010 2

2( , ) siniE t

nnx t x

L Lπ −⎡ ⎤⎛ ⎞Ψ = ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

e⎣ ⎦

---- (4) ---- (4)

V(x)

x 0 L

V E

I II III

Quantum tunneling Quantum tunneling Consider the case where Consider the case where a particle of mass m is attempting to penetrate a potential barrier of height V in the positive x-direction from section I to section III.

a particle of mass m is attempting to penetrate a potential barrier of height V in the positive x-direction from section I to section III. For section I: For section I:

2 2

2( ) ( )I ImEf x f x

x∂

= −∂

where ( ) (incident wave) (reflected wave) ikx ikxIf x Ae Be−= +

For section II:

2

2

2 2

1 ( )2 ( )

2 ( ) ( )( )

IIII

IIII

f x V Em f x x

m V E f xf xx

∂− +

∂−∂

=∂

=

where ( ) +Dnx nxIIf x Ce e−=

For section III:

2 2

2( ) ( )III IIImEf x f x

x∂

= −∂

where ( ) (transmitted wave) ikxIIIf x Fe=

Applying the boundary conditions,

(0) (0)I IIf f= '(0) '(0)I IIf f= ( ) ( )II IIIf L f L= '( ) '( )II IIIf L f L= we get, A B C DAik Bik Cn Dn+ = +− = −

+D -Dn

nL nL ikL

nL nL ikL

Ce e FeCne e ikFe

−

−

=

= Rearranging,

1 1 1 00

0 00 0

nL nL ikL

nL nL ikL

B Aik n n C ikA

e e e Dne ne ike F

−

−

−⎛ ⎞⎜ ⎟ ⎢ ⎥ ⎢ ⎥−⎜ ⎟ ⎢ ⎥ ⎢ ⎥=⎜ ⎟ ⎢ ⎥ ⎢ ⎥−⎜ ⎟ ⎢ ⎥ ⎢ ⎥− −⎝ ⎠

⎡ ⎤ ⎡ ⎤

⎣ ⎦ ⎣ ⎦

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Using Gaussian elimination,

1 1 1 00

0 00 0

1 1 1 0

1 00000

1 1 1 0

20 00000 0 2 ( )

nL nL ikL

nL nL ikL

nL nL ikL

nL nL ikL

nL nL ikL

nL ikL

Aik n n ikA

e e ene ne ike

Ani ni Ak k

ne ne nene ne ike

Ak in k in A

k ke e e

ne n ik e

−

−

−

−

−

−

−⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟

− −⎝ ⎠−⎛ ⎞⎜ ⎟−⎜ ⎟

= ⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟− −⎝ ⎠−⎛ ⎞⎜ ⎟− +⎜ ⎟

= ⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟− −⎝ ⎠

*

* *

2

* *

2* *

**

*

* 2* 2

1 1 1 020 1 0

0 1 00 0 2 0

1 1 1 020 1 0

20 0

0 0 2 0

1 1 1 0

0 1 0 2

0 0 1 2

0 0 12

nL iZL

nL ikL

nL iZL

nL ikL

iZL

nLnL

iZ L

AZ AkZ Z

e ene iZe

AZ AkZ Z

Z Ake eZ Z

ne iZe

AZ

AkZZe Z

AkZ e Z

Z eiZe

n

−

−

−

−

−−

−⎛ ⎞⎜ ⎟⎜ ⎟

= ⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟− −⎝ ⎠−⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟−

− −⎜ ⎟⎜ ⎟⎜ ⎟− −⎝ ⎠−

= −−

−−

*

* *

*

* 2 * 2

*

* 2* 2

0

1 1 1 0

20 1 0

20 0 1

20 0 0

2

ikL

nL nL

iZ L iZL

nLnL

Z

AZ AkZ Z

e Z AkZ e Z Z e Z

AkiZe e ZZ e Zn Z e Z

− −

−−

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠−⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= − −⎜ ⎟

− −⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟−−⎝ ⎠

where Z k ni= +

CKW2010 4

Hence,

*

*

*

* 2 * 2

* 2 *

* 2 ( ) ( ) *

( )

* 2 2 *

( )

* 2

22

4( ) 2

4( ) 2

4( ) 2

4( 2 )

iZ L iZL

nL nL

nL iZ L iZL

nL ki n L ki n L

n ki L

nL nL

n ki L

nL

iZe e Z AkFn Z e Z Z e Z

F nkA Z e Z iZe ne Z

nkZ e Z iZe ne Z

nkeZ e Z iZe nZ

nkeZ iZ n ZiZe

− −

−

− + −

−

−

−

⎡ ⎤+ =⎢ ⎥

− −⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥− +⎣ ⎦

⎡ ⎤= ⎢ ⎥− +⎣ ⎦⎡ ⎤

= ⎢ ⎥− +⎣ ⎦⎡

=+ −

( )

* 2 2

( )

2 2 2

( )

2 2 2 2

( )

2 2 2 2

4( ) ( )

4( ) ( )

4( 2 ) ( 2 )

4( )( 1) 2 ( 1)

n ki L

nL

n ki L

nL

n ki L

nL

n ki L

nL nL

nkeZ ki n i k ni e

nkiek ni k ni e

nkiek nki n k nki n e

nkiek n e nki e

−

−

−

−

⎤⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥+ − +⎣ ⎦⎡ ⎤

= ⎢ ⎥− − + +⎣ ⎦⎡ ⎤

= ⎢ ⎥− − − + + −⎣ ⎦⎡ ⎤

= ⎢ ⎥− − + +⎣ ⎦

2

2 2 22

2 2 2 2 2 2 2 2

2 2

2 2 2 2 2

2

2 2 2 2 2 2

16| |(2 ) (1 ) ( ) ( 1)

16(2 ) ( ) ( ) ( )

(2 )(2 ) cosh ( ) ( ) sinh ( )

nL

nL nL

nL nL nL nL

F n k eA nk e k n e

n knk e e k n e e

nknk nL k n nL

− −

⎡ ⎤= ⎢ ⎥+ + − −⎣ ⎦

⎡ ⎤= ⎢ ⎥+ + − −⎣ ⎦⎡ ⎤

= ⎢ ⎥+ −⎣ ⎦

2

2

2 2 2 2 2 2 2 2 2 2

2

2 2 2 2 2

22 22

(2 )(2 ) cosh ( ) (2 ) sinh ( ) (2 ) sinh ( ) ( ) sinh ( )

(2 )(2 ) ( ) sinh ( )

1

1 sinh ( )2

nknk nL nk nL nk nL k n nL

nknk k n nL

k n nLnk

⎡ ⎤= ⎢ ⎥− + + −⎣ ⎦⎡ ⎤

= ⎢ ⎥+ +⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎛ ⎞+⎢ ⎥+ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

CKW2010 5

Simplifying,

12

2 2 2

2 2

12

2

1

2

2 2 ( )

1 s2 2 ( )2

1 sinh ( )2 ( )

11 sinh ( )4 1

mE m V E

T nmE m V E

V nLE V E

nLE EV V

inh ( )L

−

−

−

⎡ ⎤⎛ ⎞−⎢ ⎥+⎜ ⎟⎢ ⎥⎜ ⎟= +⎢ ⎥⎜ − ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥= + ⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟= +

⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦

where 1EV⎛ ⎞ <⎜ ⎟⎝ ⎠

Approximating,

1

2 2 21 11 (44 1

nL nLT eE EV V

1 )e

−

−

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎛ ⎞⎢ ⎥⎜ ⎟= + −⎜ ⎟⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎝ ⎠−⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦

2 216 1 (1 )nL nLE E e eV V

2− − −⎛ ⎞⎛ ⎞≈ − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

for large L.

216 1 nLE E eV V

−⎛ ⎞⎛ ⎞≈ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

2nLe−∝ where 2

2 2

2 ( ) 8 ( )m V E m V Enh

π− −= =

learning outcome (s) in 9646 A-level syllabus.

CKW2010 6