Download 3-D Schrodinger`s Equation, Particle inside a 3

Introduction
Gomen-nasai:
Have not finished grading midterm II
Problem I issues:
p = mv =
h
l
Is the deBroglie wavelength for a matter
wave, p=mv not valid for a photon
Instead use relativistic kinematics, E= c p to solve
for the photon momentum.
Problem II issues
æ 1
1 ö
= Rç 2 - 2 ÷
l
èn L n Uø
1
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For hydrogen atom, need to
change R for a hydrogen-like
atom (but this is not even
necessary for part b), where the
energies Ea and Eb are given).
Introduction (cont’d)
Problem III issues: Where is it most likely to find the
particle for n=1 ?
No complicated
calculations
using the
Heisenberg
uncertainty
principle.
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Goals for Chapter 41 (Atomic Structure)
•
To write the Schrödinger equation for a three-dimensional problem
•
To learn how to find the wave functions and energies for a particle in
a three-dimensional box
•
To examine the full quantum-mechanical description of the hydrogen
atom
•
To learn about quantization of orbital angular momentum; will
examine how an external magnetic field affects the orbital motion of
an atom’s electrons
•
To learn about the intrinsic angular momentum (spin) of the electron
•
To understand how the exclusion principle affects the structure of
many-electron atoms
•
To study how the x-ray spectra of atoms indicate the structure of these
atoms
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Introduction
• The Bohr model, in which an atom’s electrons orbit its nucleus like
planets around the sun, is inconsistent with the wave nature of
matter. A correct treatment uses quantum mechanics and the threedimensional Schrödinger equation.
• To describe atoms with more than one electron, we also need to
understand electron spin and the Pauli exclusion principle. These
ideas explain why atoms
that differ by just one
electron (like lithium with
three electrons per atom
and helium with two
electrons per atom) can
be dramatically different
in their chemistry.
Li
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He
The Schrödinger equation in 3-D
• Electrons in an atom can move in all three dimensions of space. If
a particle of mass m moves in the presence of a potential energy
function U(x, y, z), the Schrödinger equation for the particle’s
wave function ψ(x, y, z, t) is
• This is a direct extension of the one-dimensional Schrödinger
equation.
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The Schrödinger equation in 3-D: Stationary states
• If a particle of mass m has a definite energy E, its wave function
Ψ(x, y, z, t) is a product of a time-independent wave function
Ψ(x, y, z) and a factor that depends on time but not position. Then
the probability distribution function |Ψ(x, y, z, t)|2 = |Ψ (x, y, z)|2
does not depend on time (stationary states).
  x, y, z, t     x, y, z  eiEt /
• The function Ψ(x, y, z) obeys the time-independent Schrödinger
equation in three dimensions:
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Particle in a three-dimensional box
• For a particle enclosed in a cubical box with sides of length L
(see Figure below), three quantum numbers nX, nY, and nZ label
the stationary states (states of definite energy).
• The three states shown here are degenerate: Although they have
different values of nX, nY, and nZ, they have the same energy E.
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Particle in a 3-D box: Separation of Variables
Important technique for partial differential equations.
y (x, y,z) = X(x)Y (y)Z(z)
Now insert in here
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Particle in a 3-D box: Separation of Variables (cont’d)
Now divide by X(x)Y(y)Z(z)
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Particle in a 3-D box:Boundary conditions/results
On the walls, X, Y, Z must be zero 
X(x)=0 at x=0 and x=L;
Y(y)=0 at y=0 and y=L;
Z(z)=0 at z=0 and z=L
Y (y) = C
nY p y
)
ny
Y sin(
L
nX p x
nZ p z
Xnx (x) = C X sin(
)
Znz (z) = CZ sin(
)
L
L
nXp x
nY p y
nZ p z
y nx ,ny ,nz (x, yz) = C sin(
)sin(
)sin(
)
L
L
L
(nx = 1,2, 3...;ny = 1,2, 3...;nZ = 1,2, 3...)
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Application of this mathematics for EM waves
Standing EM waves inside a rectangular
microwave oven
On the walls, X, Y, Z must be zero 
X(x)=0 at x=0 and x=L;
Y(y)=0 at y=0 and y=L;
Z(z)=0 at z=0 and z=L
nXp x
nY p y
nZ p z
y nx ,ny ,nz (x, yz) = C sin(
)sin(
)sin(
)
L
L
L
(nx = 1,2, 3...;ny = 1,2, 3...;nZ = 1,2, 3...)
Question: Why do the n’s start from one ?
Is nx=0 a solution ?
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Clicker question on 3-D particle in a box
A particle in a cubical box is in a state of definite energy.
The probability distribution function for this state
A. oscillates in time, with a frequency that depends on
the size of the box.
B. oscillates in time, with a frequency that does not
depend on the size of the box.
C. varies with time, but the variation is not a simple
oscillation.
D. does not vary with time.
E. answer depends on the particular state of definite
energy
Copyright © 2012 Pearson Education Inc.
Clicker question 3-D particle in a box
A particle in a cubical box is in a state of definite energy.
The probability distribution function for this state
A. oscillates in time, with a frequency that depends on
the size of the box.
B. oscillates in time, with a frequency that does not
depend on the size of the box.
C. varies with time, but the variation is not a simple
oscillation.
D. does not vary with time.
E. answer depends on the particular state of definite
energy
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Clicker question 3-D particle in a box
A particle is in a cubical box with sides at x = 0, x = L, y = 0, y
= L, z = 0, and z = L.
When the particle is in the state nX = 2, nY = 1, nZ = 1, at
which positions is there zero probability of finding the
particle?
A. on the plane x = L/2
B. on the plane y = L/2
C. on the plane z = L/2
D. more than one of A., B., and C.
E. none of A., B., or C.
Copyright © 2012 Pearson Education Inc.
Clicker question 3-D particle in a box
A particle is in a cubical box with sides at x = 0, x = L, y = 0, y
= L, z = 0, and z = L.
When the particle is in the state nX = 2, nY = 1, nZ = 1, at
which positions is there zero probability of finding the
particle?
A. on the plane x = L/2
B. on the plane y = L/2
C. on the plane z = L/2
D. more than one of A., B., and C.
E. none of A., B., or C.
nXp x
nY p y
nZ p z
y nx ,ny ,nz (x, yz) = C sin(
)sin(
)sin(
)
L
L
L
(nx = 1,2, 3...;ny = 1,2, 3...;nZ = 1,2, 3...)
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The hydrogen atom: Quantum numbers
• The Schrödinger equation for
the hydrogen atom is best
solved using coordinates (r, θ,
ϕ) rather than (x, y, z) (see
Figure at right).
• The stationary states are
labeled by three quantum
numbers: n (which describes
the energy), l (which
describes orbital angular
momentum), and ml (which
describes the z-component of
orbital angular momentum).
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The hydrogen atom: Schrodinger Equation
Use this potential in the Schrodinger
Equation
2
-1 e
U(r) =
4pe 0 r
Use the separation of
variables technique and
spherical coordinates
y (r,q ,f ) = R(r)Q(q )F(f )
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The hydrogen atom: 3-D Schrodinger Equation
y (r,q ,f ) = R(r)Q(q )F(f )
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The hydrogen atom: Results
-13.6eV
En =
n2
This result agrees
with the Bohr model
!
Here l=0,1,2,….n-1
This result does not agree
with the Bohr model.
Question: Why ? What
happens for n =1 ?
Here m=0,±1, ±2,…. ±l
The Bohr model does not include
this part at all.
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