Lecture 11' Bound States, Particle in a Box

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Lecture 11. Bound States, Particle in a Box

• Outline
• Midterm I results
• The time-independent Schrödinger Equation
• Energy Eigenfunctions
• Bound Systems
• Infinite Potential Well

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Results of Midterm 1
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Time-Independent Potential
The Schrödinger equation plays a role analogous
to Newtons 2nd Law given suitable initial
conditions (typically, ?(x,0)), it determines
?(x,t) for all future time.
- differential equation, no general solving
method other than trial and error
If the potential is independent of time
we can separate variables
the time-independent S. Eq.
- the operator of total energy, Hamiltonian
- the solution of the t-independent S.Eq. that
corresponds to E (the so-called energy
wavefunction). This is the wavefunction that a
particle would have if we knew with certainty
that its energy has a specific value E.
The solutions correspond to the states of
definite total energy
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Importance of Energy Eigenfunctions

• The energy eigenfunctions correspond to the
  stationary (t-independent) states

- the probability density is t-independent
Every expectation value is constant in time
(show this for momentum).

• The general solution of the t-dependent S. Eq.
  is a linear combination of energy eigenfunctions

The energy eigenfunctions form a complete basis
in a complex Hilbert space extension of the
methods of vector algebra to infinite-dimensional
spaces.
Given a t-independent potential U(x) and the
starting wave function ?(x,0). The corresponding
t-independent S.Eq. yields an infinite collection
of energy wavefunctions, each with its associated
value of the energy eigenvalues (separation
constant)
?i - energy eigenfunctions
It is always possible to match the initial state
by appropriate choice of the constants cn
Thus, once youve solved the t-independent S.
Eq., youre essentially done!
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Importance of Energy Eigenfunctions (contd)
Example A particle starts out in a linear
combination of just two stationary states
The wave function at subsequent times
If one measures an energy for an ensemble of
identical quantum systems in such a state, the
results of measurements will provide either E1 or
E2
- the probability density oscillates, this is
certainly not a stationary state!
- interference of probabilities

• If a particle is in a state with a well-defined
  energy E, its easy to decide whether it is
  bound or not.

• In many cases, an initial wavefunction that is
  not an energy eigenfunction will fairly quickly
  evolve so that it becomes an energy
  eigenfunction. For example, an electron whose
  wavefunction does change shape with time will
  spontaneously radiate electromagnetic energy, in
  contrast to the electrons whose wavefunctions are
  stationary. Therefore an electron whose initial
  wavefunction is not an energy eigenfunction will
  radiate photons until its wavefunction becomes an
  energy eigenfunction. This process generally
  takes on the order of 10-8 s .

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Bound Systems
A bound system is any system of interacting
particles where the nature of the interactions
between the particles keeps their relative
separation limited. Classical example the solar
system.
In general, the problem is very difficult.
Simplification motion of a single particle that
moves in a fixed potential energy field U(x)
created by the other particles in the system. A
good approximation when the mass of the particle
is small compared to the total mass of the system
(think heavy nucleus ? light electron).
Classically allowed region
Classical bound system
Classically forbidden region
The Newtonian mechanics puts no constraints on
the particles energy (continuous spectrum).
The probability to find a classical object is
proportional to (velocity)-1. The classical
probability is peaked at the turning points,
where v0.
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The Infinite Square Well
- a particle in the potential is completely free,
except at the two ends where an infinite force
prevents it from escaping
Outside the well
- the probability of finding the particle 0
Inside the well
- the harmonic oscillator equation
- constants A and B are fixed by boundary
conditions
General solution
Continuity of the wave function
Thus,
n quantum number (1D motion is characterized by
a single q.n., for 2D motion we need two quantum
numbers, etc.)
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Wavefunctions and Energy Quantization
- the pre-factor from normalization condition
Allowed energies of a bound system are quantized
In the process of measurement, the particle
wavefunction must collapse to one of the energy
wavefunctions, and En are the only possible
results of such a measurement.
The ground state n1
Excited states ngt1
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Signatures of the Quantum Behavior

• Energy quantization - only discrete energy levels
  are allowed in a steady state.
• Zero-point energy - the lowest possible energy
  level of the particle, called the zero-point
  energy, is nonzero.
• Spatial nodes - In contrast to classical
  mechanics, where the probability of finding the
  particle is uniform throughout the well, the
  probability distribution for a quantum particle
  in a stationary bound state is NOT uniform there
  are nodes where the probability is ZERO!
• due to interference effects caused by the
  wave-like character of quantum particles
• (compare with the standing waves on a string of
  length L)

In general, (n-1) nodes for a wavefunction of the
energy level n.
(For a potential with infinite walls, there are
two additional zeros at the wall location.)
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Side-by-side comparison classic vs. quantum
The difference is most dramatic for low energy
states (e.g., for the ground state).
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Recapitulation
Ground state and the uncertainty principle
- the estimate is 40 times smaller than the
exact result the contribution of the regions
where I?I2 is small is exaggerated by the U.P.
Better accuracy if we assume ?xL/2.
Standing de Broglie waves
- in agreement with the exact solution of the
S.Eq.
The time-averaged momentum in the stationary
states is zero! E.g., for the ground state
(see next slide)
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Operators of Position and Momentum
For a particle in state ?, the expectation value
of x is
Because of the time dependence of ?, the
expectation value of x will change with time
Note that the velocity of the expectation value
of x is not the same thing as the velocity of the
particle. Well postulate that the expectation
value of the velocity is equal to the time
derivative of the expectation value of position
All classical dynamical variables can be
expressed in terms of x and p. For example
kinetic energy is
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Example
In is becoming possible to construct
nanostructures that can confine electrons to a
tiny strip of metal (or semiconductor). This
so-called quantum dot can be modelled as a box
with infinitely high walls. Evaluate the energy
of an electron in such a box that is 6 nm long.
Even at room temperature (kBT0.026eV) most of
these quantum dots will be in their ground state
What photon energy is required to excite the
electron in the ground state to the next
available energy level (n2)?
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Example
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Example
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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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Sketching a wavefunction (contd)
Sketch the wavefunction with n5
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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
The wavefunction of a particle in a 2D well with
nx4 and ny4
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Man-made quantum box
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HW 5
Homework 5 Beiser Ch. 5, Problems 4, 5, 9,
10, 15, 17, 26, 28, 30, 34