Lecture 12: 02/13/09

II1,3 Plausibility Argument leading to Schrödinger's Equa tion:

II1,4 Physical Significance of the Wave Function ΨΨΨΨ(x,t):

A. 0

B. 1

C. infinite

D. Something else

( )∫∞

∞−

=Ψ ?, 2dxtxFor a proper wave function,

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5am

plitu

de

x [arb. units]

real(Ψ)imag(Ψ)

ΨΨ*

0 0.2 0.4 0.6 0.8 1-2

-1

0

1

2

ampl

itude

x [arb. units]

real(Ψ)imag(Ψ)

ΨΨ*

II1,5 Expectation Values

A. x=0

B. Something between –L/2 and +L/2

C. Something between – ∞∞∞∞ and +∞∞∞∞D. Something else

A particle is associated with the following wave function:

for –L/2<x<L/2

elsewhere

If the position x of the particle would be measured, the result would be:

( ) tieLx

Ltx ωπ −=Ψ )cos(

2,

( ) 0, =Ψ tx

A. <x> = 0

B. <x> = L/2

C. <x> = -L/2

D. <x> ≈≈≈≈ L/4

E. Something else

A particle is associated with the following wave function:

for –L/2<x<L/2

elsewhere

What is the expectation value of the position < x>?

( ) tieLx

Ltx ωπ −=Ψ )cos(

2,

( ) 0, =Ψ tx

Symmetry of probability density about x=0! -> <x> =0