Download Chapter 6: Basics of wave mechanics A bit of terminology and

Chapter 6: Basics of wave mechanics
A bit of terminology and formalism
The standard basic examples of free and bound states
6.1 Basics and definitions
Developed in the 1920’ies by Louis de Broglie, Erwin Schrödinger, Max Born,
Werner Heisenberg, Paul Dirac, Wolfgang Pauli and many others.
We have to formulate and solve the Schrödinger equation for a given problem.
First discuss stationary problems: HfÝrÞ = EfÝrÞ
2
¥ 42 + E
with H = ? 2m
pot Ýr Þ
the ”Hamilton Operator” (operator of total energy).
to solve this Eigenwert problem
describe suitable boundary conditions (Randbedingungen)
which reduce the infinite amount of solutions of the differential equation
to the physically meaningfulwave functions f L ÝrÞ
belongingto well defined energy eigenvaluesE L
(e.g. n in the Bohr model)
with a set of so called quantum numbers L
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Basics and definitions continued.
HfÝrÞ = EfÝrÞ with H =
2 2
¥
? 2m 4
+ E pot Ýr Þ
To illustrate the procedure we first treat the 1-D case:
2 d 2 fÝxÞ
¥
? 2m
dx2
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+ E pot ÝxÞfÝxÞ = EfÝxÞ
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6.1.1 Free particle
2 d 2 fÝxÞ
¥
? 2m
dx2
+ E pot ÝxÞfÝxÞ = EfÝxÞ
⇒
2 d 2 fÝxÞ
¥
? 2m
dx2
= EfÝxÞ
a trival example: E > 0 , no force acting on the particle (no potential) with
k2
=
d 2 fÝxÞ
dx2
p2
¥2
= 2mE
:
2
¥
+ k 2 fÝxÞ =0
solution fÝxÞ = expÝ+ikxÞ and expÝ?ikxÞ (wave traveling in + and - x direction)
wÝxÞ = |fÝxÞ | 2 = 1 is dependentof x!
All energies ÝE > 0Þ are physically meaningfull
no boundary restrictions: particle can move from x = ?K to +K
The most generalsolution is: fÝxÞ =A expÝikxÞ + B expÝikyÞ
(does not describe a particle moving freely into one direction )
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6.1.2 Unbound particle (energy > 0) in a potential
In a potential the kinetic energy of the particle changes
E kin = E ? E pot ÝxÞ
accordingly we rewrite the Schrödinger equation:
2
¥ 2 d fÝxÞ
2m dx2
d 2 fÝxÞ
dx2
+ ÝE ? E pot ÝxÞÞfÝxÞ = 0
+ k 2 ÝxÞfÝxÞ = 0 with k 2 ÝxÞ =
2mÝE?E pot ÝxÞÞ
¥2
This is not yet a solution of the problem
- but a way of writing it a so that we can guess the general behaviour of the
fÝxÞ i expÝ?ikÝxÞxÞ
solution:
Note aside: A realistic, approximativesolution for slowly varying E pot ÝxÞ
2m
X ÝE ? E pot ÝxÞ dx
is the so called WKB approximation: fÝxÞ i exp ?i
¥
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6.1.3 Particle in a box (Kastenpotenzial)
2
¥ 2 d fÝxÞ
2m dx2
+ ÝE ? E pot ÝxÞÞfÝxÞ = 0
E pot ÝxÞ =
Epot
K if
x< 0
0
0 ² x< a
if
K if
a² x
solution:
0
x
0
fÝxÞ =
a
x< 0
if
expݱikxÞ if
0 ² x< a
0
a² x
if
generalsolution: fÝxÞ = A expÝ+ikxÞ + B expÝ?ikxÞ
!
fÝxÞ = 0 at x = 0 and x = +a
boundary conditions: continuity (Stetigkeit):
ì standing waves with nodes on boundary
fÝxÞ = A sinÝkxÞ
ka = n^ (quantum number n = 1, 2, 3. . . . . Þ
f n ÝxÞ = A sinÝ n^
a xÞ
hence the energies (eigenvalues): E =
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¥ 2k 2
2m
=
FU - Physik III - WS 2000/2001 I.V. Hertel
n 2¥ 2^2
2ma 2
=
n2
h2
8ma 2
= n2 E1
6
6.2 General considerations for bound states
Bound states: E < 0
basically something like ”standing waves”
schrödinger (Albert)
E
0
x
note: for E ? E pot < 0 (classically forbidden region)
k becomes imaginary
ì expݱikxÞ í expݱnxÞ
with n =
2m
¥
E pot ? E
E
0
x
we have to fullfill boundary conditions
on the left and right:
for 2nd order diff. equation that implies
continuity and (in general) differentiability (Stetigkeit und Differenzierbarkeit)
ì only a set of well defined ”stationary” states with eigenenergies is allowed
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6.3 Observables - 6.3.1 Basic definitions
This is an important term in quantum mechanics:
Observables are all parameters (variables) which can be measured.
e.g. position of a particle x, y, z
momentum of a particle p x, p y, p z
energy E, angular momentum L 2
projection of L an axis L z etc.
Observables are described by operators , say G.
There are special states of a quantum system - called eigenstates d of an
operator G (also eigenvectors, eigenfunctions), in which this observable
assumes well defined values, so called eigenvaluesg.
Eigenstates and eigenvaluesof an observableare determinedby solving its
eigenvalueequation:
Gd L = g L d L
Complete set of eigenstates d L ì any state f of the system is a linear
superposition: f = > aL d L Orthonormality: Ýd L1 |d L2 Þ = X X X d DL d L2 d 3 r = N L1L 2
1
The most famous exampleof such an eigenvalueequation is the Schrödinger
equation Hf = Ef
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6.3.2 Simultaneous measurement of several observables
Note: not all observablescan be measured simultaneously
or more precisely: .. have simultaneously well defined values
e. g. not x and p x (see uncertainty relation AxAp x > h/2)
But we may very well measure exactly and simultaneously x and p z etc.
Quantum mechanics defines these interrelation in a systematic manner
At present we only need to discuss how we describe a measurement
Any physical situation can be described by a suitable state (wavefunction),
or by an ensembleof states.
The probablity to find a particle at r in the volume element d 3 r is
wÝrÞd 3 r = |fÝrÞ | 2 d 3 r =|fÝr, S, jÞ | 2 r 2 dr sin S dS dj = |fÝx, y, zÞ | 2 dx dy dz
normalisation: X|fÝrÞ | 2 d 3 r = 1 also written: X|fÝrÞ | 2 d 3 r = XfÝrÞf D ÝrÞd 3 r =Ýf|f Þ
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6.3.3 Measurement of an eigenvalue.
If of a system is in an Eigenstate d L of the observableG
defined by Gd L = g L d L
and we make a measurementof G
the experimentwill us allways give the value g L
Example: The Eigenvalues of the Hamiltonian H of an atomic hydrogen atom
are the energies E n = ?
E0
2n 2
(as in the Born model).
If we know that an H-atom is in the first excited state
(quantum number n = 2Þ
it is described by an Eigenstate f 2 ÝrÞ
and any measurementof the energy of that state
will give the identical value E 2 = ?
E0
2n 2
But what, if the state under consideration
is not an Eigenstate of the observable to be measured?
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6.3.4 Measurement of an expectation value
Clearly, we can measure any physically meaningful quantity G
for any state f - as often as we want.
However, the measurementof observablesG for which
the state f under considerationis not an eigenstate(Gf ® gfÞ
does not give unique values in the measurements.
each measurement can give another value - obeying statistial laws
Quantum mechanics makes predictions about the probability
to measure a certain observableG:
it predicts the average value < G > of the quantity under consideration
called expectation value of G
ÖGÝrÞ × = X d 3 r f D ÝrÞGÝrÞfÝrÞ
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more general: Ö G × = Ýf|Gf Þ
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6.3.5 Example and illustration
determine the average value of x from a known wavefunction,
(in this case the result is trivial x ¯ 0 )
say fÝxÞ = sin x/x
Systematically: we divide the x ? scale in boxes of size Ax
w(x)
count, how often a value of x is measured
Ni
say: in a total of N measurements
we find the particle N 1 times at x1
N 2 times at x2 , N 3 times at x3 etc.
Ax
x
The average value of x is then
x x x
1
x =
2
3
1 ÝN x
1 1
N
+ N 2 x2 + N 3 x3 +. . . Þ =
> N i xi
N
=
> wÝxi Þxi
N
í X|fÝxÞ | 2 xdx
expectation value of G : ÖGÝrÞ × = X d 3 r f D ÝrÞGÝrÞfÝrÞ = Ýf |Gf Þ
ÖGÝx, y, zÞ × = X X X f D Ýx, y, zÞGÝx, y, zÞfÝx, y, zÞdxdydz
in polar coordinates: ÖGÝr, S, jÞ × = X X X f D Ýr, S, jÞGÝr, S, jÞfÝr, S, jÞr 2 dr sin S dS dj
for an eigenstated this reduces to: G = Ýd|Gd Þ = Ýd|gd Þ = gÝd|d Þ = g
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6.3.6 Terminology, Orthonormality
Summary: Observablesare all parameters which can be measured
Eigenstates and eigenvalues of an observable are
determinedby solving its eigenvalueequationGd L = g L d L
subject to certain boundary conditions
the solutions form a complete basis set which usually is orthonormalised:
X X X d 3 r d DL1 ÝrÞd L2 ÝrÞ = d L1 |d L2 = N L1L2
Any (pure) state f of the system can be expressed as a linear superposition
of the Eigenstates (to any operator) f = > aL d L
Eigenstates (and any pure state) are also written as |d L Þ and Ýd L |
they are typically characterised by a set
of quantum numbers L
bra
ket
hence, simplified , one writes : |LÞ and ÝL|
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6.4 Potential step
d 2 fÝxÞ
dx2
+ 2m
ßVÝxÞ ? E àfÝxÞ = 0
2
¥
E
V(x)
VÝxÞ =
0
if
?K < x < 0
V0
if
0 ² x< K
general solution fÝxÞ =A expÝ+ikxÞ + B expÝ?ikxÞ
í +x
è ?x
k =
2mE
¥
2mÝE?V 0 Þ
¥
if
?K < x < 0
if
0 ² x< K
V0
x
Distinguish two cases:
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6.4.1 Potential step 0<E<V0
2mE
¥
k =
i
fÝxÞ =
= real
2mÝV 0 ?EÞ
¥
if
= in
if
?K < x < 0
V(x)
0 ² x< K
V0
E
A expÝ+ikxÞ + B expÝ?ikxÞ if ?K < x < 0
C expÝ?nxÞ + D expÝ+nxÞ if
x
0 ² x< K
no exponential growth
and at x = 0 continuity of the function and first derivative:
ì B = C?A
fÝ0Þ ì A + B = C
df
dx
x=0
ì ikA ? ikB = ?nC
2ik A ? A
B = ik?n
Ý
Þ
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ì ikA ? ikÝC ? A Þ = ?nC
= 2ik?ik+n
A
ik?n
2ik A
ì C = ik?n
Ý
Þ
= ik+n
A
ik?n
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Potential step 0<E<V0 continued (1)
f = A
expÝ+ikxÞ + ik+n
expÝ?ikxÞ if ?K < x < 0
ik?n
2ik
Ýik?n Þ
expÝ?nxÞ
if
0 ² x< K
V(x)
V0
E
x
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Potential step 0<E<V0 continued (2)
f = A
expÝ+ikxÞ + ik+n
expÝ?ikxÞ if ?K < x < 0
ik?n
2ik
Ýik?n Þ
expÝ?nxÞ
if
ψ 2, V(x)
V0
0 ² x< K
E
x
the matter wave enters into the classically forbidden region
penetration depth 1/n =
¥
2m ÝV 0 ?E Þ
the reflected wave is: B expÝ?ikxÞ = ik+n
A expÝ?ikxÞ
ik?n
2
f
|
|
We define the current of a travellingwave (or wave packet): j = v|f | 2 = ¥k
m
In the present case:
2 = ¥k ik+n A
B
j refl = ¥k
|
|
m ik?n
m
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2
2 ik+n ?ik+n = ¥k |A | 2 = j
= ¥k
A
|
|
in
m
m
ik?n ?ik?n
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