Download Schroedinger equation Basic postulates of quantum mechanics

Schroedinger equation
Basic postulates of quantum mechanics.
Operators: Hermitian operators, commutators
State function:
eigenfunctions of hermitian operators-> normalization, orthogonality
completeness
eigenvalues and expectation values of operators
Time independent Schroedinger equation and stationary states.
Probability current.
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Schroedinger equation.
Schroedinger equation is a wave equation, which links time evolution of
the wave function of the state to the Hamiltonian of the state.
For most of systems Hamiltonian
“represents” total energy of the system T+V= kinetic +potential.
Hamiltonian is defined also classically, and equations of motions for
classical systems can be written using derivatives of the Hamiltonian.
Classically there is no need for a concept of the wave function of the state,
as any state can be totally specified by giving momentum and positions of all
particles.
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
- First I remind you about a flat wave . This is the wave function describing
a free particle.
- I will show that the flat wave is a solution of the free Schroedinger
equation.
- It useful to “test” operators and properties of wave functions
on a flat wave to understand what they really mean.
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
A non-relativistic particle has the following Hamiltonian-> energy:
2
p
H =[
V  r  ]
2m
where kinetic energy is
m v 2 p 2
Ek=
=
2
2m
we know that a free particle ( propagating in a place without potential) can
be described by a flat wave or a combination of flat waves- wave packet
 m  r , t = A e
i
p r −Et
h
 r , t =2  h
−3
2
h
h =E ,=h / p , ℏ=
2
where
∫ p e
i
p r −Et
h
3
d p
we note that:
−3
i
p r −Et
∂  r , t 
−i
3
=2  h 2  ∫ p  E e h
d p
∂t
h
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Thus
−3
i
∂  r , t 
p h p r −Et 3
2 −i
=2  h  ∫ p 
e
d p
∂t
h
2m
2
Now remind ourselves Laplace operator
∇ = ∂ 2 ∂ 2 ∂ 2
∂x ∂y ∂z
2
2
2
∂  r , t 
2
∂x
2
−3
2
=−2  ℏ 
and note that for example :
1
 p  p x e
2 ∫
2
i
p r −Et
ℏ
3
d p
ℏ
i
p r −Et
1
2
2 ℏ
2
∇   r , t =−2  ℏ  2 ∫ p  p e
d3 p
ℏ
thus
2
−3
∂
h
2
ih
=[−
∇ ]
∂t
2m
2
basic principles 20/01
This is Schroedinger
equation for free particle
Anna Lipniacka www.ift.uib.no/~lipniack/
We can also note that differentiating over x (for example):
−3
i
p r −Et
h ∂  r , t 
3
2
h
=−2  h ∫ p  p x e
d p
i
∂x
and define momentum operators
h ∂
h ∂
h ∂
p x ≝
, p y ≝
, p z ≝
i ∂x
i ∂y
i ∂z
2
2

∂

p

∂
h
2

ih
= H =[
]
ih
=[−
∇ ]
∂t
2m
∂t
2m
Natural extension to the situation with potential ( non-free particle)
2

∂ 
p
ih
= H =[
V  r ]
∂t
2m
∂
h2 2
ih
=[−
∇ V  r ]
∂t
2m
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Postulates of Quantum Mechanics
A ) For every “classical” observable a linear operator. It can depend on momentum and position
operators. Momentum operator (x) is proportional to derivative over x
F  q q .. q , p p .. p : q =q , p = ℏ ∂
1, 2,
n
1, 2,
n
n
n
n
i ∂ qn
B) A system is fully described by a wave-function which fulfills wave equation, Schroedinger
2
for non-relativistic Hamiltonian
C)
∂ 
p
ih
= H =[
V  r ]
∂t
2m
expectation value of an operator F
< F >=∫  F  d 
*
corresponds to the expectation ( mean value) of the measurement result of the variable F
taken over a big number of independent measurements (this is more difficult
then it sounds)
D) the only possible results of single measurements of the variable F are the eigenvalues
of the operator F
F = f 
f = constant
(example of spin ½ , z projections )
(after a measurement the system “collapses” to
the state with well defined variable f)
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
What's new here ?
You have heard this before on “kvantefysikk og statistik mekanikk”
The difference is that now we are trying to write our wave function in
a more general way. It can be a function of space variables, or momentum
or perhaps even more general variables ( and time).
In practice we will start with space representation, then we will discuss
momentum representation, and then other- general Dirac representation
of the sate and vector representation ( matrix mechanics)
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Ad C.
< F >=∫  F  d 
*
d =dq 1 .... d q n
In general we integrate over the set variables the wave function of the system is
defined on, and normalized on. In “space representation” of quantum mechanics
we discuss right now this is space coordinates. For example an expectation
value of a position of a particle will be:
< r t >=∫∫∫   r , t  r   r , t  d r
*
3
< r t  >=∫∫∫ r ∣  r , t ∣ d r
2
3
what about momentum ?
1=∫∫∫ ∣  r , t ∣ d r
2
“proper normalization”
3
This is probability density to find the state in location r
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Hermitian operators.
The expectation values of an operator representing any real variable must be real
*
*


< F >=∫  F  d  = ∫   F  d =< F >
*
Definition of hermitian operator: The operator F is hermitian if :
*


∫  F  2 d =∫  2  F 1  d 
*
1
If the operator is hermitian its expectation value is REAL- that is a requirement for
operators associated with observables
Definition of adjoined operator to F

*


∫  F  2 d =∫  2  F 1  d 
*
1
Hermitian operators are self-adjoined meaning
basic principles 20/01



: F =F
example, check p_x
operator.
Anna Lipniacka www.ift.uib.no/~lipniack/
Ad hermitian, prove of hermeticity condition
*


< F >=∫  F  d =∫   F  d 
*
(alpha, any constant number- take real functions PSI1, PSI2
ia
* 
ia
ia
ia 
*


e


F

e


d
=

e


F

e
F


d
∫ 1
∫ 1
2
1
2
2
1
2
* 
ia
* 
−ia
* 
* 


F

d
e


F

e


F




F 2 d  =
∫ 1
∫ 1
∫ 2
1
2
1 ∫
2
 * d 
=   F  * d e−ia   F  * eia   F  *    F 
∫
1
∫
1
1
2
∫
2
1
∫
2
2
*


∫  F  2 d =∫  2  F 1  d 
*
1
*


∫  F 1 d =∫ 1  F  2  d 
*
2
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Commutator of operators
Commutators: are important! for example variables who's operators commute
can be measured simultaneously for the system.
 ,B
 ]= A
 B−
 B
 A

[A
Lets check
[ x , p x ]= x p x − p x x 
ℏ ∂ ℏ ∂ x 
x p x − p x x =x
−
i ∂x i ∂x
ℏ ∂ ℏ ∂ ℏ
=x
− x
− =ℏ i 
i ∂ x i ∂ x i x position variable, and p_x do
not commute.
[ x , p x ]=i ℏ What does that mean for
subsequent measurements of px and x ?
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Eigenfunctions and eigenvalues:
The result of an operator working on a function is usually a different function . If there exist
a set of functions that :
F  n = f n  n
we call them eigenfunctions of the operator F, and fn are eigenvalues (constants)
The set of eigenvalues ( or spectrum of the operator) can be discrete , continuos or mixed.
Example of continuos- energy operator (Hamiltonian) for a free particle, discrete= hamiltonia
for harmonic oscillator, mixed- hamiltonian for hydrogen atom.
ℏ ∂
p x =
i ∂x
ℏ ∂
ifx
 f = f  f   f =const ∗exp  
i ∂x
ℏ
That's the form of eigenfunctions of momentum operator. They are not quadraticaly
integrable and have to be normalized in a different way.
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Expectation value vs. eigenvalue of an operator.
< F >=∫  F  d 
*
The expectation ( mean value) of the measurement of the variable F
taken over a big number of independent measurements ( in practice over large number
of identically prepared states- an ensamble ).
But the only possible results of measurements of the variable F are the eigenvalues
of the operator F
(example of spin ½ , z projections )
F = f 
If the system is in the state described by the eigenfunction of the operator ,the expectation
value of the measurement is equal to the eigenvalue. As the eigenvalue
is the only possible measurement results, that is what we will always get !
* 
*

< F >=∫  F  d =∫ n F n d = f n∫ n n = f n
*
for the eigenstate number “n”
This proof is straightforward for normalized quadratic-integrable eigenfunctions,
can be also proved for different type of normalization
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Orthogonality of eigenfunctions
For a hermetic operator, eigenfunctions corresponding to different eigenvalues
are orthogonal
*
*
∫  n F  m d =∫  m  F  n  d 
*
*
*
f m∫  n  m d = f n∫  m  n d 
*
 f m − f n ∫  n  m d =0
if
f m ≠ f n then
Dirac notation:
∫
*
n
 m d =〈 n∣m〉
∫
 m d =0
Scalar product
∫
〈 n∣F m〉=〈 F n∣m〉
basic principles 20/01
*
n
*
n
F  m d =〈 n∣F m〉
Hermitian operator in Dirac notation
Anna Lipniacka www.ift.uib.no/~lipniack/
We try to normalize our set of eigenfunctions of an operator – typically
choosing a constant to multiply the functions with.
∫
*
n
 m d = nm
0 for n = m , orthogonality
1 for n=m , normalization
Normalization of eigenfunctions with continuous spectrum of
eigenvalues has to be a bit different- functions are not “quadratically
integrable”
∫  f '  f d = f ' − f =< f' | f>
*
ifx
 f  x =c∗exp 

ℏ Example
∞
1
∫  '  x f  x  d x= 2  ℏ −∞
∫e
*
f
C must be
basic principles 20/01
c=
1
2  ℏ
ix
Dirac delta
momentum eigenfunctions, here just 1-dim
 f − f '
ℏ
dx= f ' − f 
to get the normalization right
Anna Lipniacka www.ift.uib.no/~lipniack/
More about Dirac delta function:
Dirac delta function :
∞
∞
−∞
−∞
∫ f  x  x −x '  dx = f  x ' 
∫ f  x  x  dx= f 0
∞
∫ 1∗ x  dx =1
properties :
−∞
possible representation:
∞
1

 x =lim   0    x =lim 
exp
ixy
−
y

dy=lim


∣
∣
∫
 0
2
2
2  −∞
 x  
∞
1
 x =
∫ exp ixy  dy
2  −∞
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
   x =

 x  
2
2
=0.01
=0.1
=0.5
x
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Completeness
We assume that every “reasonable” , quadratically integrable function can
be expressed as a linear combination of eigenfunctions of a hermitian operator
F
g=∑ c i  i  | g >=∑ c i |i >=∑ <i | g> |i >
i
i
i
for orthonormal set of eigenfunctions the coefficients are a scalar product
of the function in question and appropriate eigenfunction
∫  n g d =∑ c i∫  n i d =c n  c n=∫  n g d =<n | g>
*
*
*
i
For eigenfunctions with continuous spectrum of eigenvalues
we can represent a function g in the following way : (eigenfunctions have to form a
complete basis set)
g=∫ c  f  f df
 | g >=∫ df c  f | f >=∫ df <f| g> | f >
∫  f '  f d = f ' − f =< f' | f>
*
then we have :
c  f =∫  g d =< f | g>
*
f
we must also have :
basic principles 20/01
∫
*
f
 r '  f  r  d f = r ' −r 
Anna Lipniacka www.ift.uib.no/~lipniack/
Completeness, continuation: Lets now consider that our functions
are normalized in the normal “space”, so
∫  f '  f d =∫  f '  r  f  r  d r = f ' x − f x  f ' y − f y  f ' z− f z 
*
*
3
etc..
c n =∫  n  r '  g r '  d r '
g r =∑ c i  i  r 
*
i
g r =∫ g r '  ∑  i  r '  i  r  d r '
*
i
3
3
∑  i  r ' i  r = r −r ' 
*
i
3-dim Dirac
delta
we must in analogy
have for continuous
spectrum:
basic principles 20/01
∫  f  r '  f  r  d f = r ' −r 
*
Anna Lipniacka www.ift.uib.no/~lipniack/
What is the interpretation of expansion coefficients cn ? We see that modulus
squared of an expansion coefficient will correspond to a probability
to measure certain eigenvalue: Lets take spectrum of eigenfunctions
of given operator F and expand a function of state (g) into it :
g=∑ c i  i
and check what is the expectation value of the operator
F for the state described by g
i
*
* 

< F >=∫ g F g d =∫  ∑ c n  n  F  ∑ c i  i 
*
n
i
< F >=∫  ∑ c n  n  ∑ f i c i  i =∑ ∑ c n * c i f i∫  n  i d 
*
*
n
*
i
n
i
< F >=∑ ∑ c n * c i f i i,n =∑ ∣c i∣ f i
2
n
i
i
but we know that from interpretation of expectation value we must have
< F >=∑ P i f i
i
basic principles 20/01
where P_i is the probability that the value f_i
will be measured.
Anna Lipniacka www.ift.uib.no/~lipniack/
Thus we have.
The probability that measuring observable associated with F on a state
described by a wave function “g” will give a result f_i is the following:
P i =∣c i∣ =∣∫  g d ∣
2
F  i = f i  i
2
*
i
where
For continuos spectrum of F we can prove in analogy that:
〈 F 〉=∫ f ∣c  f ∣ df
2
The probability that measuring observable associated with F on a state
described by a wave function “g” will give a result beetween f and f+df is
P  f  df =∣c  f ∣ df =∣∫  g d ∣ df
2
basic principles 20/01
*
f
2
where
F  f = f  f
Anna Lipniacka www.ift.uib.no/~lipniack/
Stationary states:
If the Hamiltonian does not contain time explicitly we can try to separate the solutions
in to time dependent part and coordinates dependent part. We obtain that there is
new constant involved proportional to the time derivative of the time dependent
function divided by the function itself. We call it ENERGY. We obtain time-independent
Shroedinger equation for the part which does not depend on time.
t , r ≝  r T t 
dT t / dt
iℏ
=const ≝E
T t 
   r =E   r 
H
−i Et / ℏ
T =C e
∂ T t 
h2 2
ih
  r =[−
∇  r V  r   r ]T t 
∂t
2m
What “stationary” means in practice ? Expectation values of operators
do not depend on time. ( for normal operators which do not contain
time derivatives ) Prove it !
basic principles 20/01
Anna Lipniacka www.ift.uib.no/~lipniack/
Probability current :
Probability interpretation for the particle wave function: modulus square of it is a probability
to find a particle in a given place (probability density).
THUS: Integral over space has to give 1. However locally spacial probability density can change
with time. We can define the probability current, useful when discussing movement of particles
∂  r , t  ∂  
∂
ℏ
ℏ
* ∂
*
2
2
*
*
*
=
=

=i
 ∇ − ∇  =−i
∇  ∇  − ∇ 
∂t
∂t
∂t
∂t
2m
2m
*
*
change of probability density with time (at a given place)
is related to the out-flow of the current.
∂
ℏ
=−i
∇  ∇ * − * ∇ =−∇ j
∂t
2m
j =i ℏ  ∇  * − * ∇ =ℜ * i ℏ ∇ 
2m
m
∂
h
2
ih
=[−
∇ V  r ]
∂t
2m
2
basic principles 20/01
∂
h
2
*
ih
=[−
∇ V  r  ]
∂t
2m
*
2
Anna Lipniacka www.ift.uib.no/~lipniack/