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Unformatted text preview: I. The Double Slit
Experiment I. The Double Slit
Experiment
x 1. Electron gun
or laser x=0
2. (Any QM “particles”) Let P(x) be the probability that an electron will arrive at point x.
What will we see?
(ignoring single slit diffraction) 2 I. The Double Slit
Experiment A. The Classical Picture
P(x) = P1(x)+P2(x) P1(x): Probability of passing through
slit 1 and reaching position x.
P2(x): Probability of passing through
slit 2 and reaching position x.
Only
slit 1
open Only
slit 2
open Both
slits
open + =
Bransden
and Joachain
page 53
3 I. The Double Slit
Experiment B. Reality
P(x) ≠ P1(x)+P2(x) Pattern characteristic of wave interference
This is true even for single electrons let through
one at a time
Individual electrons detected at speciﬁc points
(particlelike) but diffraction pattern builds up
over time (wave like) Only
slit 1
open Only
slit 2
open Both
slits
open + =
Bransden
and Joachain
page 53
4 I. The Double Slit
Experiment B. Reality Pattern characteristic of wave interference
This is true even for single electrons let through
one at a time
Individual electrons detected at speciﬁc points
(particlelike) but diffraction pattern builds up
over time (wave like) Bransden
and Joachain
page 54
5 I. The Double Slit
Experiment C. The Quantum Picture For waves, intensity is described by the
square of an amplitude.
We need something like that now, but
something that accounts for the particle
aspect.
instead of intensity instead of amplitude like a wave, it can be,
in general, complex P (r, t) ∝ ψ (r, t) = ψ (r, t)ψ (r, t)
2 Probability of detecting a particle at r,t
(real positive number) Probability amplitude (wave
function, state function)
6 ∗ I. The Double Slit
Experiment
C. The Quantum Picture P (r, t) ∝ ψ (r, t) = ψ (r, t)ψ (r, t)
2 2 Only slit 1 open 2 ∗ Only slit 2 open P1 ∝ ψ1 
P2 ∝ ψ2  P ∝ ψ1 + ψ2 2 Both slits open 7 I. The Double Slit
Experiment D. More Formally The probability of ﬁnding a particle within
the volume element dr = dxdydz centered
about the point r = (x,y,z) at a time t is P (r, t)dr = ψ (r, t) dr
2 so that P (r, t) ∝ ψ (r, t)2 = ψ (r, t)ψ ∗ (r, t)
Probability density Theory has a statistical nature: one predicts the
probability of a result, not a speciﬁc result.
8 I. The Double Slit
Experiment
D. More Formally The particle must be detectable somewhere so
the total probability should be 1 (100% chance
of ﬁnding it somewhere). P (r, t)dr = ψ (r, t)2 dr = 1 9 I. The Double Slit
Experiment E. The Superposition Principle What is the probability density for the double
slit experiment?
Instead of adding probabilities (classical), one
adds wavefunctions.
Superposition: If ψ1 describes a possible state of a
system, and ψ2 describes another possible state of the system,
then any linear combination ψ = c1 ψ1 + c2 ψ2 is also a
possible state of the system.
constant are, in general, complex too
10 I. The Double Slit
Experiment F. A Look at the Postulates of Quantum Physics We’ve seen two of the postulates of quantum
physics. From Bransden and Joachain:
1. “To an ensemble of physical systems one can, in certain
cases, associate a wave function or state function which
contains all the information that can be known about the
ensemble. This function is in general complex; it may be
multiplied by an arbitrary complex number without
altering its physical signiﬁcance.” 11 I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of
state function which contains all the information that can be known about the ensemble. This
function is in general complex; it may be multiplied by an arbitrary complex number without
altering its physical signiﬁcance.” 2. The superposition principle. This is also
a possible
state ψ = c1 ψ1 + c2 ψ2 + · · ·
one possible
state 12 another
possible
state I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of
state function which contains all the information that can be known about the ensemble. This
function is in general complex; it may be multiplied by an arbitrary complex number without
altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · What are the other postulates?
3. With every dynamical variable is associated a linear
operator. 13 I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of
state function which contains all the information that can be known about the ensemble. This
function is in general complex; it may be multiplied by an arbitrary complex number without
altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator. 4. The only result of a precise measurement of the
dynamical variable A is one of the eigenvalues an of the
linear operator A associated with A. 14 I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all
the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an
arbitrary complex number without altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator.
4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear
operator A associated with A. 5. If a series of measurements is made of a dynamical
variable A on an ensemble of systems described by the
wavefunction ψ, the expectation value, or average
value of this dynamical variable is
drψ ∗ (r)Aψ (r)
ψ Aψ
A =
=
ψ ψ
drψ ∗ (r)ψ (r)
15 I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all
the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an
arbitrary complex number without altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator.
4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear
operator A associated with A.
5. If a series of measurements is made of a dynamical variable A on an
drψ ∗ (r)Aψ (r)
ψ Aψ
A =
=
ensemble of systems described by the wavefunction ψ, the
ψ ψ
drψ ∗ (r)ψ (r)
expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can
be expressed as a linear combination of the eigenfunctions
of A, where A is the operator associated with a dynamical
variable.
16 I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all
the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an
arbitrary complex number without altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator.
4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear
operator A associated with A.
5. If a series of measurements is made of a dynamical variable A on an
drψ ∗ (r)Aψ (r)
ψ Aψ
A =
=
ensemble of systems described by the wavefunction ψ, the
ψ ψ
drψ ∗ (r)ψ (r)
expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A,
where A is the operator associated with a dynamical variable. 7. The time evolution of the wavefunction of a
system is determined by the time dependent
Schrödinger equation ∂ψ (r, t)
i
= Hψ (r, t)
∂t
17 I. The Double Slit
Experiment
F. A Look at the Postulates of Quantum Physics
1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all
the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an
arbitrary complex number without altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · 3. With every dynamical variable is associated a linear operator.
4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear
operator A associated with A.
5. If a series of measurements is made of a dynamical variable A on an
drψ ∗ (r)Aψ (r)
ψ Aψ
A =
=
ensemble of systems described by the wavefunction ψ, the
ψ ψ
drψ ∗ (r)ψ (r)
expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A,
where A is the operator associated with a dynamical variable. 7. The time evolution of the wavefunction of a system is determined by the time dependent Schrödinger
equation: i ∂ψ (r, t)
= Hψ (r, t)
∂t where H is the Hamiltonian, or total energy operator, of the system. 18 a
r
b ce
ea
g sp
lt
A er
r
a
I. The Double Slit
Experiment F. A Look at the Postulates of Quantum Physics 1. “To an ensemble of physical systems one can, in certain cases, associate a wave function of state function which contains all
the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an
arbitrary complex number without altering its physical signiﬁcance.”
2. The superposition principle. ψ = c1 ψ1 + c2 ψ2 + · · · e
in
L 3. With every dynamical variable is associated a linear operator. lb
i
H 4. The only result of a precise measurement of the dynamical variable A is one of the eigenvalues an of the linear
operator A associated with A. a
n
o
5. If a series of measurements is made of a dynamical variable A on an
drψ ∗ (r)Aψ (r)
ψ Aψ
A =
=
ensemble of systems described by the wavefunction ψ, the
ψ ψ
drψ ∗ (r)ψ (r)
expectation value, or average value of this dynamical variable is: 6. A wavefunction representing any dynamical state can be expressed as a linear combination of the eigenfunctions of A,
where A is the operator associated with a dynamical variable. 7. The time evolution of the wavefunction of a system is determined by the time dependent Schrödinger
equation: i ∂ψ (r, t)
= Hψ (r, t)
∂t where H is the Hamiltonian, or total energy operator, of the system. 19 We need to learn this
formalism of quantum
mechanics ...
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This note was uploaded on 09/28/2011 for the course PHYSICS 334 taught by Professor Russel during the Spring '11 term at Waterloo.
 Spring '11
 Russel
 Diffraction

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