Zeropoint Energy

1
Zero-point Energy

• Minimum energy corresponds to n1
• n0 gt ?n(x)0 for all x
• gt P(x)0 gt no electron in the well
• zero-point energy
• never at rest!
• Uncertainty principle if ?xL/?, then
  ?px gt h/(?x) ? h /L gt E gt (?px
  )2/2m ? 2 h 2/2mL2
  h2/(8mL2)

2
Problem

• An electron is trapped in an infinite well which
  is 250 pm wide and is in its ground state. How
  much energy must it absorb to jump up to the
  state with n4?
• Solution standing waves gt n(?/2)L
• Ep2/2m h2/2m ?2 n2h2/8mL2
• E4-E1 (h2/8mL2)(42 -1) 15(6.63x10-34)2/8(9
  .11x10-31)(250x10-12)290.3 eV

3
Problem

• An electron is trapped in an infinite well that
  is 100 pm wide and is in its ground state. What
  is the probability that you can detect the
  electron in an interval of width ?x5.0 pm
  centered at x (a) 25 pm (b)50 pm (c)90 pm ?
• P(x) ?(x)2 is probability/unit length
• P(x)dx probability that electron is located
  in interval dx at x
• L100 x 10-12 m ?(x)(2/L)1/2 sin(?x/L)
• P(x) ?x (2/L)sin2(?x/L) ?x (1/10) sin2(?x/L)
• a) P(x) ?x .1 sin2(?/4) .05
• b) P(x) ?x .1 sin2(?/2) .1
• c) P(x) ?x .1 sin2(.9?) .0095

4
Problem

• A particle is confined to an infinite potential
  well of width L. If the particle is in its ground
  state, what is the probability that it will be
  found between (a) x0 and xL/3 (b) xL/3 and
  x2L/3 (c) x2L/3 and xL?
• If ?x is not small, we must integrate!

5
(No Transcript)
6
Solution

• Total probability of finding the particle between
  xa and xb is

7
Solution (contd)

• Since cos(2y) cos2(y)-sin2(y)1-2sin2(y)
• we have sin2(y) (1/2)(1 - cos(2y))

8
Solution(contd)

• For a0, bL we have L/L -
  (1/2?)sin(2?)-sin(0) 1
• (a) a0, bL/3 1/3 - (1/2?)sin(2?/3)-sin(0)
  .195
• (b) aL/3, b2L/3 1/3 - (1/2?)sin(4?/3)-si
  n(2?/3) .61
• (c) a2L/3, bL 1/3 - (1/2?)sin(2?)-sin(4?
  /3) .195

9
Infinite Potential Well
?
?
Nodes at ends gt standing waves ?(x)0
outside
10
Electron in a Finite Well

• Only 3 levels with E lt U0
• states with energies E gt U0 are not confined
• all energies for E gt U0 are allowed since the
  electron has enough kinetic energy to escape to
  infinity

Quantization of energy
11
No longer a node at x0 and xL
Electron has small probability of penetrating
the walls
Area under each curve is 1
12
Harmonic Oscillator Potential
P(x) ?0 everywhere