Kuliah Fisika Modern 2009 – fisika itu sunatullah 03-schroedinger_eq
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
Schr¨odinger Equation
an introduction
Febdian Rusydi
PHOTON – University of Airlangga
15 Mei 2009
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
1
Mechanism of Motion
review so far
dynamics
2
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
QUIZ #1 DIMULAI DALAM 5 MENIT KE DEPAN
Buat jarak antaranda
Kosongkan meja-kursi Anda kecuali sehelai kertas dan alat
tulis dan kalkulator.
Tidak ada peringatan; kertas jawaban Anda langsung diambil
jika Anda membuat gerakan mencurigakan. Pembeladaan diri
dapat dilakukan setelah kelas usai.
Tidak diperkenankan meminjam alat tulis dan kalkulator.
Waktu 20 menit tanpa perpanjangan.
SEMUA BENTUK KECURANGAN DIGANJAR NILAI UAS
NOL.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
Quiz (20 minutes)
At time t = 0 a particle is represented by the wave function
(
A (a2 − x 2 )
if − a ≤ x ≤ +a ,
Ψ(0, x) =
0
otherwise.
1
Determine the normalization constant A.
2
Sketch Ψ(0, x) as a function of x.
3
What is the expectation value of x (at time t = 0).
4
Find is the expectation value of p (at time t = 0). Note: You
cannot get it from m dhxi/dt, why?
5
Check whether the particle still obeys the Heisenberg’s
uncertainty principle.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
foundation principle
1
2
The very basic principle
probability
the very basic principle
⇓
|Ψ(t, x)|2 = PDF
Born’s statistical
interpretation
The basic assumptions
de Broglie hypothesis
the first assumption
⇓
Febdian Rusydi
wave-particle duality
they behave symmetrically
⇓
particle has wavelength
λ = ~/p
Heisenberg’s Uncertainty
Principle
the second assumption
⇓
σ x σp > ~
limitation of measurement
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
EOM wave is Wave Equation → theory of wave for wave.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
EOM wave is Wave Equation → theory of wave for wave.
Theory of particle for light → photon.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
EOM wave is Wave Equation → theory of wave for wave.
Theory of particle for light → photon.
Theory of wave for particle → ?
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
EOM must be symmetrical
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
dotting your ℑ’s and crossing your h’s!
~2 2
−
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
Laplacian operator (cartesian):
∇2
Febdian Rusydi
∂2
∂2
∂2
+
+
∂x 2 ∂y 2 ∂z 2
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
It is a different form of the conservation energy principle.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
It is a different form of the conservation energy principle.
It was published by Ewrin Schr¨
odinger in 1925; he later won
the Noble Prize at 1933 for that.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
It is a different form of the conservation energy principle.
It was published by Ewrin Schr¨
odinger in 1925; he later won
the Noble Prize at 1933 for that.
A year before, in 1924, Werner Heisenberg published a
seemingly different theory to explain quantum physics.
Schr¨
odinger succeed to show that both equations are equally
the same. Their both equations found the ”quantum
mechanics”.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
~2 2
∇ +V
−
2m
Febdian Rusydi
Ψ = ℑ~
∂
Ψ
∂t
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
~2 2
∇ +V
−
2m
Ψ = ℑ~
∂
Ψ
∂t
~ = m ~a, in the
It was analogue to Newton’s dynamics law, F
sense that (1) it is EOM for non-relativistic case and (2) it
relates the time and space derivatives of the wave functions.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
~2 2
∇ +V
−
2m
Ψ = ℑ~
∂
Ψ
∂t
~ = m ~a, in the
It was analogue to Newton’s dynamics law, F
sense that (1) it is EOM for non-relativistic case and (2) it
relates the time and space derivatives of the wave functions.
The kinetic energy form,
T =−
~2 ∂ 2 Ψ
2 m ∂x 2
is impossible to be derived from anywhere... it came from
Schr¨
odinger’s head and was the real masterpiece of his work.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
position, momentum, kinetic
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
Hamiltonian: H = T + V .
p2
+ V (x)
2m
~
in QM: x → xˆ ; p → pˆ = ∇ ;
ℑ
in CM: H =
ˆ = pˆ
T
2m
To find the value → calculate the expectation value of the
operator.
Z +∞
ˆ x , pˆ) Ψ
hQi =
Ψ∗ Q(ˆ
−∞
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
please read
Thursday: section 3.1 – 3.8
Friday: section 3.9 – 3.10
Febdian Rusydi
Schr¨
odinger Equation
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
Schr¨odinger Equation
an introduction
Febdian Rusydi
PHOTON – University of Airlangga
15 Mei 2009
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
1
Mechanism of Motion
review so far
dynamics
2
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
QUIZ #1 DIMULAI DALAM 5 MENIT KE DEPAN
Buat jarak antaranda
Kosongkan meja-kursi Anda kecuali sehelai kertas dan alat
tulis dan kalkulator.
Tidak ada peringatan; kertas jawaban Anda langsung diambil
jika Anda membuat gerakan mencurigakan. Pembeladaan diri
dapat dilakukan setelah kelas usai.
Tidak diperkenankan meminjam alat tulis dan kalkulator.
Waktu 20 menit tanpa perpanjangan.
SEMUA BENTUK KECURANGAN DIGANJAR NILAI UAS
NOL.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
Quiz (20 minutes)
At time t = 0 a particle is represented by the wave function
(
A (a2 − x 2 )
if − a ≤ x ≤ +a ,
Ψ(0, x) =
0
otherwise.
1
Determine the normalization constant A.
2
Sketch Ψ(0, x) as a function of x.
3
What is the expectation value of x (at time t = 0).
4
Find is the expectation value of p (at time t = 0). Note: You
cannot get it from m dhxi/dt, why?
5
Check whether the particle still obeys the Heisenberg’s
uncertainty principle.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
foundation principle
1
2
The very basic principle
probability
the very basic principle
⇓
|Ψ(t, x)|2 = PDF
Born’s statistical
interpretation
The basic assumptions
de Broglie hypothesis
the first assumption
⇓
Febdian Rusydi
wave-particle duality
they behave symmetrically
⇓
particle has wavelength
λ = ~/p
Heisenberg’s Uncertainty
Principle
the second assumption
⇓
σ x σp > ~
limitation of measurement
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
EOM wave is Wave Equation → theory of wave for wave.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
EOM wave is Wave Equation → theory of wave for wave.
Theory of particle for light → photon.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
consequence of de Broglie hypothesis
Definition (mechanism)
Mechanism is the way that something works like the way it does.
Mathematically mechanism is stated as EOM. Since EOM is for
dynamic object, a working only happen if the object is dynamics.
Dynamic objects: particle and wave.
EOM particle is Newton 2nd law → theory of particle for
particle.
EOM wave is Wave Equation → theory of wave for wave.
Theory of particle for light → photon.
Theory of wave for particle → ?
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
review so far
dynamics
EOM must be symmetrical
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
dotting your ℑ’s and crossing your h’s!
~2 2
−
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
Laplacian operator (cartesian):
∇2
Febdian Rusydi
∂2
∂2
∂2
+
+
∂x 2 ∂y 2 ∂z 2
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
It is a different form of the conservation energy principle.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
It is a different form of the conservation energy principle.
It was published by Ewrin Schr¨
odinger in 1925; he later won
the Noble Prize at 1933 for that.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
It is a different form of the conservation energy principle.
It was published by Ewrin Schr¨
odinger in 1925; he later won
the Noble Prize at 1933 for that.
A year before, in 1924, Werner Heisenberg published a
seemingly different theory to explain quantum physics.
Schr¨
odinger succeed to show that both equations are equally
the same. Their both equations found the ”quantum
mechanics”.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
~2 2
∇ +V
−
2m
Febdian Rusydi
Ψ = ℑ~
∂
Ψ
∂t
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
~2 2
∇ +V
−
2m
Ψ = ℑ~
∂
Ψ
∂t
~ = m ~a, in the
It was analogue to Newton’s dynamics law, F
sense that (1) it is EOM for non-relativistic case and (2) it
relates the time and space derivatives of the wave functions.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
commentary on the Schr¨odinger equation
~2 2
∇ +V
−
2m
Ψ = ℑ~
∂
Ψ
∂t
~ = m ~a, in the
It was analogue to Newton’s dynamics law, F
sense that (1) it is EOM for non-relativistic case and (2) it
relates the time and space derivatives of the wave functions.
The kinetic energy form,
T =−
~2 ∂ 2 Ψ
2 m ∂x 2
is impossible to be derived from anywhere... it came from
Schr¨
odinger’s head and was the real masterpiece of his work.
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
position, momentum, kinetic
−
~2 2
∇ +V
2m
Ψ = ℑ~
∂
Ψ
∂t
Hamiltonian: H = T + V .
p2
+ V (x)
2m
~
in QM: x → xˆ ; p → pˆ = ∇ ;
ℑ
in CM: H =
ˆ = pˆ
T
2m
To find the value → calculate the expectation value of the
operator.
Z +∞
ˆ x , pˆ) Ψ
hQi =
Ψ∗ Q(ˆ
−∞
Febdian Rusydi
Schr¨
odinger Equation
Outline
Mechanism of Motion
Dotting Your ℑ’s and Crossing Your h’s!
wave equation for electron
operators
next
please read
Thursday: section 3.1 – 3.8
Friday: section 3.9 – 3.10
Febdian Rusydi
Schr¨
odinger Equation