Lecture 19. Overview

1
Lecture 19. Overview

• Wavefunctions sketching, normalization,
  probability density, expectation values.
• Particle in a box, harmonic oscillator
• Tunneling
• H atom, quantum numbers, eigenfunctions,
• H atom spectrum, normal Zeeman effect

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Sketching a wavefunction
This equation shows how the second derivative of
the energy eigenfunction ?, or the curvature of
the function ?(x), is linked to the value of the
eigenfunction. Sign of E-U(x) E-U lt 0 the
function ?(x) curves upward E-U gt 0 the
function ?(x) curves downward E-U 0 the
curvature 0.

• Thus, in a classically allowed region, an energy
  eigenfunction always curves toward the horizontal
  axis (wavelike), in a classically forbidden
  region away from the horizontal axis
  (exponential-like).
• Absolute magnitude of E-U(x)
• in a classically allowed region, greater
  IE-U(x)I implies shorter wavelength, in a
  classically forbidden region steeper
  exponential tails
• the amplitude of oscillations in the
  classically allowed region is bigger when
  IE-U(x)I is smaller (compare with the probability
  of finding a slow-moving particle)

• a wavefunction of energy level n should have
  (n-1) nodes
• if the potential is symmetric with respect to
  some x, the wavefunction should be either
  symmetric or anti-symmetric with respect this x.

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Problem (Sketching Wavefunctions)
A particle is bound in the potential shown in the
Figure. Sketch the wavefunction and the
probability density corresponding to the
fifth-lowest bound state. Provide necessary
explanations.
energy
E5
U(x)
prob. density
-a
a
x
wavefunction
-a
a
x
Both plots must be even functions .
-a
a
x
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Particle in a 2D Box
Two degrees of freedom need two quantum
numbers, for x- and y-motion.
Ground state nx1 , ny1
Probability density
The wavefunction of a particle in a 2D well with
nx4 and ny4
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Particle in a 3D Box, Energy Levels
If the transition between two lowest-energy
levels is to produce a photon of 450-nm
wavelength, what should be the wells width?
nx2,ny2,nz2 nx2,ny2,nz2
nx2,ny2,nz2 - non-degenerate
nx3,ny1,nz1 nx1,ny3,nz1
nx1,ny1,nz3 triple-degenerate
nx2,ny2,nz1 nx1,ny2,nz2
nx2,ny1,nz2 triple-degenerate
nx2,ny1,nz1 nx1,ny2,nz1
nx1,ny1,nz2 triple-degenerate
nx1,ny1,nz1 - non-degenerate
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Problem (Normalization, probability)
Consider a wavefunction
where A is a constant to be determined by
normalization.
Normalize this wave function and find the
probability of the particle being between 0 and
1/?.
Normalization
Probability to find this particle in the given
interval
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Harmonic Oscillator, Expectation Values
The wavefunction of the ground state of a
particle in a harmonic potential Ukx2/2 has the
form
where m is the particle mass,
- the angular frequency.
(a) find the normalized wavefunction ?0
(a) find the expectation value of the position of
the particle in this state
(a) find the expectation values of the potential
energy and kinetic energy of the particle in this
state (express the results in terms of the
angular frequency ?).
Useful integrals
(a)
- the normalization condition
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Harmonic Oscillator, Expectation Values (contd)
- the integrand is an odd function (product of an
odd function (x) by an even function (i?i2).
(b)
(c)
Easy way out
Straightforward approach
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Tunneling
(a) What fraction of a beam of 50-eV electrons
would get through a 200-V, 0.3-nm-wide
electrostatic barrier? (Remember this is a
wide barrier). (b) What is the energy of
transmitted/reflected electrons? (c) Could the
situation depicted in the following Figure
represent a particle in a bound state? Explain.
(b) E50eV
(c) No
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Bound States, Tunneling
A particle experiences a potential energy given
by

• Make a sketch of U(x) including numerical values
  at the minima and maxima.
• What is the maximum energy the particle could
  have and yet be bound?
• Is it possible for a particle to have an energy
  greater than that in part (c) and still be
  bound for some period of time? Explain.

(b) the particle is bound if E?0
(c) the particle can be bound for some period
of time if 0 lt E ? e-4
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H Atom
For H atom, estimate
- the potential energy U is always a result of
interaction (in this particular case, an
interaction between an electron and a proton)
For a bound system electronproton,
Thus, the mass defect which corresponds to the
formation of a H atom is
Because the interaction energy is much smaller
than the electrons rest energy, we can consider
a H atom as a system of two separate particles
coupled by a (relatively weak) interaction. When
one considers strong interactions (e.g.,
interactions between quarks), this approach
doesnt work.
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H Atom
If a hydrogen atom in the 2p excited state decays
to the 1s ground state, explain how the following
properties are conserved (a) energy, (b) linear
momentum, and (c) angular momentum.
(a) energy
- the energy difference equals the energy of the
emitted photon
(b) linear momentum
- to conserve linear momentum, the magnitude of
recoil momentum of H atom must be the same as the
momentum of emitted photon (see next problem)
(c) angular momentum
- the total angular momentum is the combination
of the angular orbital momentum and the intrinsic
spin momentum. The electron spin remains the same
(?ms 0), the change in the angular orbital
momentum is compensated by the photon spin.
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Problem (H atom)
To conserve momentum, an atom emitting a photon
must recoil, meaning that not all of the energy
made available in the downward jump goes to the
photon. (a) Find a hydrogen atoms recoil
energy when it emits a photon in a n2 to n1
transition. (Note the calculation is easiest to
carry out if it is assumed that the photon
carries essentially all the transition energy
the result justifies the assumption). (b) What
fraction of the transition energy is the recoil
energy?
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H Atom
The red Balmer series line in hydrogen
(?656.5nm) is observed to split into three
different spectral lines with ??0.04nm between
two adjacent lines when placed in a magnetic
field B. What is the value of B?
?? is due to the energy splitting between two
adjacent ml states
Two adjacent lines that correspond to ?ml1
correspond to the emission of photons with
energies
( list all the states that correspond to these
energy levels)