Download Quantum Mechanics Lecture 3 Dr. Mauro Ferreira

Quantum Mechanics
Lecture 3
Dr. Mauro Ferreira
E-mail: [email protected]
Room 2.49, Lloyd Institute
Monday 10 October 2011
Postulates of Quantum Mechanics
P2:
All physically “observable” properties of a system are
represented by dynamic variables that are linear operators
Consider a linear operator Ô representing a dynamic variable
“O”. The knowledge of the wave function ψ(x) that describes the
state of a system does not provide a fully deterministic value for
the observable quantity but only a statistical distribution of
possible measurements
Monday 10 October 2011
Definition: A linear operator Ô is by definition a mathematical
entity that associates with every function ψ(x) another function
ϕ(x), the correspondence being linear.
Ô ψ = φ
Ô(λ1 ψ1 (x) + λ2 ψ2 (x)) = λ1 Ô ψ1 (x) + λ2 Ô ψ2 (x)
Examples of linear operators are:
Π̂ ψ(x, y, z) = ψ(−x, −y, −z)
(Parity operator)
∂
D̂x ψ(x, y, z) =
ψ(x, y, z)
∂x
(Differential operator)
Monday 10 October 2011
Postulates of Quantum Mechanics
P2:
All physically “observable” properties of a system are
represented by dynamic variables that are linear operators
Consider a linear operator Ô representing a dynamic variable
“O”. The knowledge of the wave function ψ(x) that describes the
state of a system does not provide a fully deterministic value for
the observable quantity but only a statistical distribution of
possible measurements
In mathematical terms, we can express the above by
!O" =
!
∞
−∞
dx ψ ∗ (x) Ô ψ(x)
The operator is “sandwiched” by the wave functions and integrated
Monday 10 October 2011
!x" =
!
∞
dx ψ (x) X̂ ψ(x)
∗
}
⇒
X̂
=
x
−∞
! ∞
ˆ
!
⇒ R = !r tic
∗
!x" =
dx ψ (x) x ψ(x)
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r (x) P̂ ψ(x)
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!p"W=
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! ∂
r
ene −∞
⇒ P̂ =
! ∞
i ∂x
!
∂ ψ(x)
∗
!p" =
dx ψ (x)
!!
ˆ
i −∞
∂x
!
⇒P = ∇
Monday 10 October 2011
}
i
Operators associated with a given observable can always be
expressed in terms of the canonical operators X and P, following
the same functional dependence found in classical mechanics.
Kinetic energy:
2
2
−!
∂
P̂
=
T̂ =
2m
2m ∂x2
2
T̂3D
−!2 2
=
∇
2m
Potential energy: V̂ (X̂) = V (x)
ˆ
!
V̂ (R) = V (!r)
ˆ
ˆ
ˆ
!
!
!
Angular momentum: L = R × P
Monday 10 October 2011
Interesting consequence:
Ô ψ = φ
Monday 10 October 2011
Mathematically, an operator applied to a function ψ
changes it to another function ϕ. Physically, a
measurement of O on a system in a state described
by ψ disturbs it, changing it to the state ϕ.
Remember ?
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Monday 10 October 2011
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Interesting consequence:
Ô ψ = φ
Mathematically, an operator applied to a function ψ
changes it to another function ϕ. Physically, a
measurement of O on a system in a state described
by ψ disturbs it, changing it to the state ϕ.
How does o
ne know wh
ether it is p
have compl
ossible to
ete simultan
eous knowl
specific pro
edge of two
perties of a
system, say
A and B ?
This can only be possible if the order of measurements of the
two properties does not matter, regardless of the state ψ the
system is in.
In other words...
Monday 10 October 2011
 B̂ ψ = B̂  ψ
 B̂ ψ = B̂  ψ
(Â B̂ − B̂ Â) ψ = 0
In operator form : Â B̂ − B̂ Â = 0̂
It is convenient to define the commutator of two
operators as
 B̂ − B̂  = [Â, B̂]
Monday 10 October 2011
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[
X̂,
P̂
]
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[X̂, P̂ ] = X̂ P̂ − P̂ X̂
! d
! d
−
x) ψ(x)
[X̂, P̂ ]ψ(x) = (x
i dx
i dx
! dψ(x)
! dψ(x)
!
[X̂, P̂ ]ψ(x) = x
−x
− ψ(x)
i dx
i dx
i
!
[X̂, P̂ ]ψ(x) = − ψ(x)
i
[X̂, P̂ ] = i !
It is simple to generalize
this to higher dimensions
[X̂, P̂x ] = i ! ; [Ŷ , P̂y ] = i ! ; [Ẑ, P̂z ] = i !
Monday 10 October 2011
It is straightforward to see that
[X̂, Ŷ ] = [X̂, Ẑ] = [Ŷ , Ẑ] = 0̂
and
[P̂x , P̂y ] = [P̂x , P̂z ] = [P̂y , P̂z ] = 0̂
Monday 10 October 2011
Properties of commutators
[Â, B̂] = −[B̂, Â]
(antisymmetry)
[Â, B̂ + Ĉ + D̂ + ...] = [Â, B̂] + [Â, Ĉ] + [Â, D̂] + ... (linearity)
[Â, B̂ Ĉ] = [Â, B̂] Ĉ + B̂ [Â, Ĉ] (distributivity)
[Â, λ] = 0 (commutation with scalars)
[Â, B̂ n ] =
n−1
!
B̂ j [Â, B̂] Ân−j−1
j=0
[Ân , B̂] =
n−1
!
j=0
Monday 10 October 2011
Ân−j−1 [Â, B̂] B̂ j
Consider the uncertainties of the observables A and B
written as operators
∆ =  − "Â#
∆B̂ = B̂ − "B̂#
∆ ψ(x) = χ(x)
∆B̂ ψ(x) = φ(x)
When applied to a state function ψ(x)
Applying the Schwarz inequality to both χ(x) and ϕ(x)
Monday 10 October 2011
Schwarz inequality
!
∞
−∞
dx ψ ∗ (x) ψ(x) ≥ 0
constant
writing ψ(x) as ψ(x) = φ(x) + λη(x)
!
∞
−∞
dx {φ φ + λφ η + λ η φ + λ λη η} ≥ 0
∗
∗
∗ ∗
∗
∗
which is certainly true for every λ. It is then also true for
!∞
∗
dx
η
φ
−∞
λ = − !∞
a slightly tedious algebra leads to
∗
dx η η
−∞
!
∞
−∞
Monday 10 October 2011
dx | φ|2
!
∞
−∞
dx | η|2 ≥ |
!
∞
−∞
dx φ∗ η|2
Consider the uncertainties of the observables A and B
written as operators
∆ =  − "Â#
∆B̂ = B̂ − "B̂#
∆ ψ(x) = χ(x)
∆B̂ ψ(x) = φ(x)
When applied to a state function ψ(x)
Applying the Schwarz inequality to both χ(x) and ϕ(x)
!(∆Â)2 " !(∆B̂)2 " ≥ |!∆ ∆B̂"|2
Further manipulation leads to
1
∆ ∆B̂ ≥ |"[Â, B̂]#|
2
Monday 10 October 2011
(General uncertainty principle)
1
∆ ∆B̂ ≥ |"[Â, B̂]#|
2
As a particular case, we have the operators momentum and
position
What’s the effect of
[X̂, P̂ ] = i !
!
⇒ ∆x ∆p ≥
2
confining a particle ?
(Standard uncertainty principle)
As a direct consequence, we see that the commutator of two
operators determine whether or not the quantities they are
n
o
i
t
associated with can be simultaneously specified. ntiza
Speaking of measurements ...
Monday 10 October 2011
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Ho ome abo
c
Quantization comes about when we impose that the allowed
values for an observable quantity O are the eigenvalues of the
operator Ô
Monday 10 October 2011
Eigenvalues and eigenvectors
When the result of applying an operator  to a function ψ is a
multiple of that function, the function is called an eigenfunction
and the multiple is called the eigenvalue of the operator.
eigenfunction
ÂΨa = aΨa
(eigenvalue equation)
eigenvalue
Eigenvalues can be either discrete or continuous
Monday 10 October 2011
Quantization comes about when we impose that the allowed
values for an observable quantity O are the eigenvalues of the
operator Ô
Quantization occurs for observables that have discrete eigenvalues
(e.g. atomic energies)
Q̂Ψqi = qi Ψqi
(i = 1, 2, 3, ...)
... but is not always the case, since eigenvalues can be continuously
distributed (e.g. particle position in a confined space)
Q̂Ψq = qΨq
Monday 10 October 2011
Example:
Suppose an electron is confined to a zero-potential region between two
impenetrable walls at x=0 and x=a and is described by the following wave
function
ψ(x) =
!
2
3πx
sin(
)
a
a
f or 0 ≤ x ≤ a
a) Calculate the expectation value of the kinetic energy
b) What is the corresponding uncertainty in the kinetic energy ?
Monday 10 October 2011