Download Quantum Computing Part 1: Quantum Mechanics Overview

Quantum Computing
Part 1: Quantum Mechanics Overview
Technological limits
Gordon Moore’s law:
• The observation made in 1965 by
Gordon Moore, co-founder of Intel,
that the number of transistors per
square inch on integrated circuits
had doubled every year since the
integrated circuit was invented.
• Moore predicted that this trend
would continue for the foreseeable
future.
• In subsequent years, the pace
slowed down a bit, but data density
has doubled approximately every
18 months, and this is the current
definition of Moore's Law, which
Moore himself has blessed.
• Most experts, including Moore
himself, expect Moore's Law to
hold for at least another two
decades.
Technological limits
• But all good things may come to an end…
• We are limited in our ability to increase
– the density and
– the speed of a computing engine.
• Reliability will also be affected
– to increase the speed we need increasingly smaller circuits
(light needs 1 ns to travel 30 cm in vacuum)
– smaller circuits  systems consisting only of a few
particles subject to Heissenberg uncertainty
Energy/operation
• If there is a minimum amount of energy dissipated to
perform an elementary operation, then to increase the
speed, thus the number of operations performed each
second, we require a liner increase of the amount of energy
dissipated by the device.
• The computer technology vintage year 2000 requires some
3 x 10-18 Joules per elementary operation.
• Even if this limit is reduced say 100-fold we shall see a 10
(ten) times increase in the amount of power needed by
devices operating at a speed 103 times larger than the sped
of today's devices.
Power dissipation, circuit density, and speed
In 1992 Ralph Merkle from Xerox PARC calculated that a
1 GHz computer operating at room temperature, with
1018 gates packed in a volume of about 1 cm3 would
dissipate 3 MW of power.
– A small city with 1,000 homes each using 3 KW
would require the same amount of power;
– A 500 MW nuclear reactor could only power some
166 such circuits.
Talking about the heat…
The heat produced by a super dense computing engine is
proportional with the number of elementary computing circuits,
thus, with the volume of the engine.
– The heat dissipated grows as the cube of the radius of the
device.
To prevent the destruction of the engine we have to remove the
heat through a surface surrounding the device. Henceforth,
– our ability to remove heat increases as the square of the
radius
– while the amount of heat increases with the cube of the size
of the computing engine.
Quantum; Quantum mechanics
• Quantum is a Latin word meaning
some quantity.
• In physics it is used with the same
meaning as the word discrete in
mathematics, i.e., some quantity or
variable that can take only sharply
defined values as opposed to a
continuously varying quantity.
• Quantum mechanics is a
mathematical model of the physical
world
I'll bet any quantum mechanic in the service
would give the rest of his life to fool around
with this gadget. - Chief Engineer Quinn,
Forbidden Planet , 1956
Heissenberg uncertainty principle
• Heisenberg uncertainty principle says we cannot determine both
the position and the momentum of a quantum particle with
arbitrary precision.
• In his Nobel prize lecture on December 11, 1954 Max Born says
about this fundamental principle of Quantum Mechanics :
``... It shows that not only the determinism of classical physics must
be abandoned, but also the naive concept of reality which looked
upon atomic particles as if they were very small grains of sand. At
every instant a grain of sand has a definite position and velocity.
This is not the case with an electron. If the position is determined
with increasing accuracy, the possibility of ascertaining its velocity
becomes less and vice versa.''
Quantum Weirdness
Quantum mechanics displays some
weird features that cannot be
duplicated by any classical system.
Can Weirdness be useful?
•Quantum Computing
•Quantum Encryption
•Quantum teleportation
•Quantum Global Positioning
Electromagnetic Weirdness
In 1867 Maxwell proposed a theory unifying
electricity, magnetism, and optics.
The theory was weird!
Light was propagated as a wave in a vacuum
- no underlying medium.
Only when Marconi invented the radio transmitter
and receiver did the weirdness become accepted as
normal, or even obvious.
Quantum Weirdness
some remarkable quotes
The quantum is the greatest mystery we've got. Never in my life was I
more up a tree than today. -John Wheeler
If someone says that he can think about quantum physics without
becoming dizzy, that shows only that he has not understood anything
whatever about it. - Niels Bohr
A picture may seem extraordinarily strange to you and after some time
not only does it not seem strange but it is impossible to find what there
was in it that was strange. - GERTRUDE STEIN concerning modern
art.
John Wheeler and friends
Niels Bohr and big money.
Gertrude Stein by Picasso
Classical versus Quantum Experiments
• Classical Experiments
 Experiment with bullets
 Experiment with waves
• Quantum Experiments
 Two slits Experiment with electrons
 Stern-Gerlach Experiment
Experiment with bullets
detector
H1
Gun
H2
wall
wall
(a)
(b)
Figure 1: Experiment with bullets
Experiment with bullets
detector
H1 is open
H2 is closed
P1(x)
H1
Gun
H2
wall
wall
(a)
(b)
Figure 1: Experiment with bullets
Experiment with bullets
detector
H1 is closed
H2 is open
H1
Gun
H2
P2(x)
wall
wall
(a)
(b)
Figure 1: Experiment with bullets
Experiment with bullets
detector
H1 is open
H2 is open
P1(x)
H1
Gun
H2
P12 (x)  12 (P1 (x)  P2 (x))
P2(x)
wall
wall
(a)
(b)
(c)
Figure 1: Experiment with bullets
Experiment with Waves
detector
H1 is closed
wave
source
H2 is closed
H1
H2
wall
wall
(a)
Figure 2: Experiments with waves
Experiment with Waves
detector
H1 is open
wave
source
H2 is closed
I1(x)
H1
H2
wall
wall
(a)
(b)
Figure 2: Experiments with waves
Experiment with Waves
detector
H1 is closed
wave
source
H2 is open
H1
H2
I2(x)
wall
wall
(a)
(b)
Figure 2: Experiments with waves
Experiment with Waves
H1 is open
detector
H2 is open
wave
source
I1(x)
H1
H2
I 12 (x)  h1 (x)  h2 (x)
I2(x)
wall
wall
(a)
(b)
(c)
Figure 2: Experiments with waves
2
This is a result of
interference
Two Slit Experiment
detector
Results intuitively
expected
P1(x)
H1
H2
source of
electrons
P12 (x)  12 (P1 (x)  P2 (x))
P2(x)
wall
wall
(a)
(b)
(c)
Figure 3: Two slit experiment
Are electrons
particles or
waves?
Two Slit Experiment
detector
Results
observed
P1(x)
H1
H2
source of
electrons
P12 (x)  ?
P2(x)
wall
wall
(a)
(b)
(c)
Figure 3: Two slit experiment
Two Slit Experiment With Observation
Now we add
light source
light
source
detector
Interference
disappeared!
P1(x)
H1
H2
source of
electrons
P12 (x)  12 (P1 (x)  P2 (x))
P2(x)
⇨ “Decoherence”
wall
wall
(a)
(b)
(c)
Figure 4: Two slit experiment with observation
Stern-Gerlach Experiment
1922 - a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into
two beams. Consistent with the possession of an intrinsic angular momentum and a magnetic moment
by individual electrons.
S
N
Stern-Gerlach experiment with spin-1/2 particles
Conclusions From the Experiments
• Limitations of classical mechanics
• Particles demonstrate wavelike behavior
• Effect of observations cannot be ignored
• Evolution and measurement must be distinguished
Can we use these phenomena
practically?
Quantum computing and
information
Quantum Weirdnesses
1. Complementarity: Every particle is a wave,
and every wave is a particle. It just depends on
how you measure it.
Leonard Mandel
classical particle
classical wave
quantum photon
Quantum Weirdnesses
2. Superposition: If |A> represents one state of the system,
and |B> represents another, then
a|A> + b|B> is also a state of the system.
Examples:
|here> + |there>
|live cat> + |dead cat>
.
Schrödinger Cat State
Quantum Weirdnesses
3. Entanglement: The quantum state of a two particle
system in which the states of the individual particles are
inseparable even though the two particles are separated.
A revolutionary approach to computing and communication
• We need to consider a revolutionary rather than an
evolutionary approach to computing.
• Quantum theory does not play only a supporting role by
prescribing the limitations of physical systems used for
computing and communication.
• Quantum properties such as
– uncertainty,
– interference, and
– entanglement
form the foundation of a new brand of theory, the quantum
information theory where computational and communication
processes rest upon fundamental physics.
Milestones in quantum physics
1900 - Max Plank presents the black body radiation theory; the quantum theory is born.
1905 - Albert Einstein develops the theory of the photoelectric effect.
1911 - Ernest Rutherford develops the planetary model of the atom.
1913 - Niels Bohr develops the quantum model of the hydrogen atom.
1923 - Louis de Broglie relates the momentum of a particle with the wavelength
1925 - Werner Heisenberg formulates the matrix quantum mechanics.
1926 - Erwin Schrodinger proposes the equation for the dynamics of the wave function.
1926 - Erwin Schrodinger and Paul Dirac show the equivalence of Heisenberg's matrix
formulation and Dirac's algebraic one with Schrodinger's wave function.
1926 - Paul Dirac and, independently, Max Born, Werner Heisenberg, and Pasqual
Jordan obtain a complete formulation of quantum dynamics.
1926 - John von Newmann introduces Hilbert spaces to quantum mechanics.
1927 - Werner Heisenberg formulates the uncertainty principle.
Milestones in computing and information theory
1936 - Alan Turing dreams up the Universal Turing Machine, UTM.
1936 - Alonzo Church publishes a paper asserting that ``every function which can
be regarded as computable can be computed by an universal computing
machine''. Church Thesis.
1946 - A report co-authored by John von Neumann outlines the von Neumann
architecture.
1948 - Claude Shannon publishes ``A Mathematical Theory of Communication’’.
1961 - Rolf Landauer decrees that computation is physical and studies heat
generation.
1973 - Charles Bennet studies the logical reversibility of computations.
1981 - Richard Feynman suggests that physical systems including quantum
systems can be simulated exactly with quantum computers.
1982 - Peter Beniof develops quantum mechanical models of Turing machines.
1984 - Charles Bennet and Gilles Brassard introduce quantum cryptography.
1985 - David Deutsch reinterprets the Church-Turing conjecture.
1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters discover quantum
teleportation.
1994 - Peter Shor develops a clever algorithm for factoring large numbers.
Deterministic versus probabilistic photon behavior
D1
D3
Incident beam of light
D5
Detector D1
D7
Reflected beam
Beam splitter
Transmitted beam
Detector D2
D2
(a)
(b)
• If we start decreasing the intensity of
the incident light we observe the
granular nature of light.
– Imagine that we send a single
photon.
– Then either detector D1 or detector
D2 will record the arrival of a
photon.
• If we repeat the experiment involving a
single photon over and over again we
observe that each one of the two
detectors records a number of events.
• Could there be hidden information,
which controls the behavior of a
photon?
– Does a photon carry a gene and one
with a ``transmit'' gene continues
and reaches detector D2 and another
with a ``reflect'' gene ends up at
D1?
The puzzling nature of light
Each detector
detects single
photons. Why?
What is a
hidden
information
that controls
this
In an attempt to
solve this puzzle
we design this
setup
D1
D3
Incident beam of light
D5
Detector D1
D7
Reflected beam
Beam splitter
Transmitted beam
Detector D2
D2
(a)
(b)
• Consider now a cascade of beam
splitters.
• As before, we send a single photon and
repeat the experiment many times and
count the number of events registered
by each detector.
The puzzling nature of light
(cont’d)
Why all
these
detectors
detect
light?
• According to our theory we expect the
first beam splitter to decide the fate of
an incoming photon;
– the photon is either reflected by the
first beam splitter or transmitted by
all of them.
• Thus, only the first and last detectors in
the chain are expected to register an
equal number of events.
• Amazingly enough, the experiment
shows that all the detectors have a
chance to register an event.
D1
D3
Incident beam of light
D5
Detector D1
D7
Reflected beam
Beam splitter
Transmitted beam
Detector D2
D2
(a)
(b)
Polarization experiment
• Polarization of light: electric field vector is confined to a
single direction. This is a quantum-mechanical property of
photons.
• Light source: Assume each photon has a random
polarization
• Filter A: Horizontally polarized
A
photons randomly
polarized
all photons
horizontally
polarized
1/2 original intensity
Polarization experiment
• Polarization of light: electric field vector is confined to a
single direction. This is a quantum-mechanical property of
photons.
• Light source: Assume each photon has a random
polarization
• Filter A: Horizontally polarized
• Filter B: Vertically polarized
A
photons randomly
polarized
B
all photons
horizontally
polarized
no light
Polarization experiment
• Polarization of light: electric field vector is confined to a single
direction. This is a quantum-mechanical property of photons.
• Light source: Assume each photon has a random polarization
• Filter A: Horizontally polarized
• Filter B: Vertically polarized
• Filter C: polarized at 45 degrees
A
C
B
photons randomly
polarized
1/8 original intensity
Polarization experiment explanation
• Photon’s polarization state can be modeled by a unit vector
pointing in the appropriate direction
• Let the basis vectors be  and 
• Any polarization can be expressed as
  a    b 
where a and b are complex numbers such that
a  b 1
2
2
Measurement postulate of quantum mechanics
– Any device measuring a quantum system has an
associated orthonormal basis with respect to which the
measurement takes place.
– Measurement of a state transforms the state into one of
these basis vectors.
– The probability that the state is measured as basis
vector u is the norm of the amplitude of the component
of the original state in the direction of u.
Example:
Let   a   b be the polarization state of a photon.
Then this state will be measured as  with probability |a|2
and as  with probability |b|2 .
These are indeed probabilities, since a  b  1 .
2
2
The measurement will also change the state to the result
of the measurement.
Its original value cannot be determined.
Back to the polarization experiment:
– Original light source produces photons with random
polarizations:
  a   b
– Thus 50% are measured as horizontal, and let through.
(Why?)
– Those photons are now in the horizontal state,
None are let through by the vertical filter.
• Now let the 45o filter be put behind the horizontal filter.
The 45o filter measures photon polarization with respect to
a new basis:
• Photons in state  will be measured as
probability 1/2. So half will get through.
with
• Likewise, filter B will let half of these through.
• Total photons getting through all three filters: (1/2)3 = 1/8.
Quantum Mechanics
Quantum mechanics
vs.
Formulated to explain the behavior
of microscopic systems.
Classical Mechanics
Formulated to explain the behavior of
macroscopic objects.
Newton’s second law:
F  ma  m d2x2
dt
Integrate twice → x(t).
Two constants of integration → two additional pieces of information required to
uniquely define the state of the system (e.g. xo and vo).
The state of the system is defined by: FORCES, POSITIONS, VELOCITIES
Knowledge of the initial state of the system can predict future states precisely.
Quantum Mechanics
Designed to describe the behavior observed for microscopic particles and systems.
Behavior:
• discrete rather than continuous energy levels
• photons, electrons and nuclei exhibit both wave and particle nature.
History:
• 1925 Werner Heisenberg, Max Born and Pascual Jordon introduced matrix-based
mathematical formalism to describe observed quantum mechanical phenomena
• 1926 Erwin Schrödinger introduced a differential equation and its solution that
equivalently describes equation observed quantum mechanical phenomena
dt 
i
  t 
dt
QM analogy to
Newton’s 2nd law
The postulates of quantum mechanics (QM)
Postulate I
For any possible state of a system, there is a function  of the coordinates of the
parts of the system and time that completely describes the system.
   ( x, y , z , t )
 Is called a wave function. For two particles system,
  ( x1 , y1 , z1 , x2 , y2 , z2 , t )
The wave function square 2 is proportional to probability. Since  may be
complex, we are interested in *, where * is the complex conjugate (i  -i) of .
The quantity *dt is proportional to the probability of finding the particles of the
system in the volume element, dt = dxdydz.
*

  dt  1
all _ space
that is the probability of finding the particle in the universe is 1  normalization condition.
The postulates of quantum mechanics (QM)
Orthogonality of two wave functions
  *dt  0
  * d t  0
Example: sinq and cosq are orthogonal functions.

 sin q cosqdq  0
0
Fourier series expansion – sin(nq) and cos(nq) orthogonal functions
The Wavefunction
Postulate I
For any possible state of a system, there is a function  of the coordinates of the
parts of the system and time that completely describes the system:
(t) = f + ig
• f and g are real functions of coordinates and time
• An abstract, complex quantity but related to physically measurable quantities
• State is dependent on coordinates (spatial and spin) and time
The time-dependent Schrödinger equation:
dt 
i
  t 
dt
QM analogy to
Newton’s 2nd law
A single integration with respect to time is required to obtain (t), so that only one
constant of integration is required to predict future states of the system.
The Wavefunction
How can we describe the state of a quantum mechanical system such as nuclear
spins?
Complex wavefunction:
(t) = (t,t) ≡ description of all knowable information about the state of the
system
Spatial and spin coordinates
(independent of time)
Time dependence
What if we want to know if the system is in a given state (t) at time t? The
probability that the system is in the state given by (t) at time t is:
probability density
P = *(t)(t) = ││2
For example, for a single particle at time t′, the wavefunction is (x,y,z,t′), and the
probability that time t′, the particle is in a given volume of space (dxdydz) is given
by:
│(x,y,z,t′)│2 dxdydz
dx
dz
dy
Complex Conjugate
Complex Conjugate *:
 = f + ig
* = f – ig
(replace i with –i)
* = (f + ig) (f – ig) = f2 – ifg + ifg – (i2)g = f2 + g2
real, non-negative
(as P should be!)
Normalization Condition
Since the system must exist in some state at time, t, if we integrate over all
coordinates of the system (t represents the generalized coordinates, which may
include spatial coordinates and spin state), the probability density is 1.
Normalization condition:
∫ *(t)(t)dt = 1
i.e. the probability of finding the particle somewhere in space is one.
The Schrödinger equation describes the evolution in time of a given system:
dt 
i
 H t 
dt
The Schrödinger Equation
dt 
i
 H t 
dt
constant,
often omitted
Hamiltonian = FORCES
The Hamiltonian, H, represents the forces acting on the system, which can be timedependent or time-independent.
The time-independent Schrödinger equation
If the Hamiltonian is time-independent, then the Schrödinger equation can be
time
timesolved by separation of variables.
indep. dep.
(t) = (t) f(t)
The Hamiltonian acts only on the generalized coordinate part of (t), (t), since H
is independent of time (i.e. f(t) acts as a “constant”). Choosing units so that ħ = 1:
df t 
i t 
  t f t 
dt
Multiplying both sides by ∫ *(t) dt:
i  *t  t  dt df t    *t  t  dt φt 
dt


=1
The time-independent Schrödinger equation
Noting that the wave function is normalized and multiplying by –i:
df t   i  * t  t dt f t 
   
dt

To simplify things, let’s define:

E   *t H t  dt
So that:
df t   iEf t 
dt
Now the goal is to solve this differential equation.
The time-independent Schrödinger equation
The solution is:
f t   e iEt
This can be seen by differentiating with respect to t:
df t 
 iEe iEt  iEf t 
dt
Now we can write:
t    t f t    t e iEt
The time-independent Schrödinger equation
Differentiating with respect to time:
dt 
  iE te iEt   iE  t 
dt
Multiplying both sides by i:
d t
i    E  t 
dt
dt 
i
 H t 
dt
Comparing to the original form of the Schrödinger equation, we get:
H  E 
Which is the time-independent Schrödinger equation. The time dependence can
be thought of as a phase factor that cancels when the probability distribution is
calculated:
f t   e iEt
f *t f t   eiEt e iEt  1
The time-independent Schrödinger equation
We have just shown that:
H  E 
H is an operator
E is an energy
 is the wave function
Probabilistic nature of QM
If  is a solution to the Schrödinger equation, then so is c (c = arbitrary
constant), and Σcii is also a solution, where each i is a possible state of the
system.
 isn’t really a physical wave. It is an abstract mathematical entity that yields
information about the state of the system.
 gives information on the probabilities for possible outcomes of measurements of
the system’s physical properties.
“Quantum mechanics … says a lot, but does not really
bring us any closer to the secrets of the Old One. I, at
any rate, am convinced that He does not throw dice.”
Albert Einstein
Operators
Postulate II
With every physical observable q there is associated an operator Q, which when
operating upon the wavefunction associated with a definite value of that observable
will yield that value times the wavefunction F, i.e. QF = qF.
H
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qmoper.html
Eigenvalues and Eigenfunctions
Eigenvalue equations:
An operator acts on a function to give another function:
Âf(x) = g(x)
Where  is an operator.
For example, an operator can be defined where â represents multiplying by a:
âf(x) = af(x)
Depending upon the operator, the new function can be very different from the
original function. However, in a special case, the new function is a multiple of the
original function:
Âf(x) = lf(x)
In this case, f(x) is said to be an eigenfunction of  with the associated eigenvalue,
l. In the case of the Schrödinger equation with a time-independent Hamiltonian, H
is the operator,  is the eigenfunction and E is the eigenvalue:
H  = E
Operators
(1) The operators are linear, which means that
O(1  2) = O1 + O2
• The linear character of the operator is related to the superposition of
states and waves reinforcing each other in the process
(2) The second property of the operators is that they are Hermitian (the 19th
century French mathematician Charles Hermite).
• Hermitian matrix is defined as the transpose of the complex conjugate (*) of a
matrix is equal to itself, i.e. (M*)T = M
x  iy 
 a

M  
c 
 x  iy
x  iy 
 a

M *  
c 
 x  iy
x  iy 
 a
M
( M * )T  
c 
 x  iy
In QM, the operator O is Hermitian if
*
*
*
F
O

d
t


O
F
dt


C. Hermite
http://commons.wikimedia.org/wiki/Image:Charles_Hermite_circa_1887.jpg
Hermitian Operators
The adjoint of an operator (A†) satisfies the equation:
*A† = l**
Hermitian operators are self-adjoint (A = A†):

*Afdt 











f*Adt *
This has several implications:
•Eigenvalues for Hermitian operators are real
•Eigenfunctions for Hermitian operators form a complete orthonormal set:
∫i*jdt = ∫j*idt = dij = Kronecker delta
•Hilbert space: a complete set of N orthonormal functions which constitutes a basis
set:
   cn  n
N
n 1
cn are complex numbers that can depend on time
Eigenvalues of QM operator
Postulate III
• The permissible values that a dynamical variable may have are those
given by OF = aF, where F is the eigenfunction of the QM operator
(Hermitian) O that corresponds to the observable whose permissible
real values are a.
• The is postulate can be stated in the form of an equation as
O
operator
F
wave function
= a
eigenvalue
F
wave function
Example Let F = e2x and O=d/dx
 dF/dx = d(e2x)/dx = 2 e2x
 F is an eigenfunction of the operator d/dx with an eigenvalue of 2.
Quantum Mechanics
Ai = lii
The result of making a measurement of A is one of the eigenvalues of A. That is,
only a limited set of outcomes are possible (discrete nature of quantum mechanics).
What is the value we might expect to measure?
Expectation Value
The expectation value is the average magnitude of a property sampled over an
ensemble of identically prepared systems. The expectation value, <A>, is the
scalar product of  and A:
<A> = ∫*Adt
If the wavefunction is an eigenfunction of the operator ( = n):
<A> = ∫*Adt = ∫n*Andt = ln∫n*ndt = ln
Expectation Value
If a system is in state (t,t), the average of any physical observable C at time t is:
<C> = ∫*Ĉdt
If one makes a large number of measurements of C with identical initial state
(t,0), then one obtains a set of values C1, C2, …, CN. The average of C is given
by the rule:
1 N
C   Ci
N i 1
A postulate of quantum mechanics is that the integral and summation above
provide the same value, which is the expectation value.
Quantum Mechanics
N
In general  ≠ n, but    ci i
i1
Hence:

A   * Adt





 N




i  1



i


 N




i  1

* 
i


  ci 

c*i A
N
N
N
N


i1 j1


i1 j1
N

j1
*A

 N



 j 1




j 





j 


c j

 N



 j 1

c j
dt

c*ic j *iA jdt
c*ic jl j
c*jc jl j 

dt
* dt
i j
noting : A j  l j j
noting :


0
* dt  dij  
i j
1

i j
i j
N
c 2j l j

j1
probability that cj is obtained in a single measurement
Quantum Mechanics
What does this mean?
When A is measured for a single member of an ensemble, the result is one of the
eigenvalues of A, but which one cannot be predicted in advance.
The result means that the eigenvalue lj will be obtained in a single measurement
with the probability of cj2.
So, for a single measurement, there are specified values of A that are possible, but
over an ensemble, the expectation value <A> can be a continuous value.
Example: Particle in a box
• Infinite potential well
• Particle mass m in a box
length L
• U(x)=0, if 0<x<L,
• U(x)=∞, if x<0 –or- x>L
• Boundary conditions on :
 (0)=0=(L)
h 2 d 2

 E
2
2m dx
d 2
2 Em
 2 
2
dx
h
V 
V 
V(x)
V 0
-a
a
Particle in a box
• Second derivative proportional to the function with
“-” sign
d 2
2 Em
 2 
2
dx
h
• Possible solutions: sin(kx) and cos(kx)
2 Em
k   2
h
2 Em p 2
k
 
h
h
l
2
 ( x)  A sin( kx)  B cos(kx)
k-wave vector
Particle in a box
 ( x)  A sin( kx)  B cos(kx)
Let’s satisfy boundary conditions
 (0)  A sin( k 0)  B cos( k 0)  B cos( k 0)  0
B0
 ( x)  A sin( kx)
 ( L)  A sin( kL)  0
kL  n
n
kn 
L
Wave vector is quantized!
Particle in a box
 n ( x)  A sin( kn x)
•
Quantum number n
k
n
L
n
kn 
L
2 Em
h

 2n2
2 Em
h
 h n
2
2
L2
2
2
2 Em
 2
h
2
hn
En 

2
2
2mL
8mL
Energy is quantized!
We are not done yet,
We don’t know A
Particle in a box
 n ( x)  A sin( kn x)
• We know for sure that the particles is
somewhere in the box
• Probability to find the particle in 0<x<L is 1:
• Unitarity condition:
L
2
|

(
x
)
|
dx  1

0
L
L
2 kL
A
1   | ( x) | dx   A sin (kx)dx 
k
0
0
2
2
2
2
sin
 ( y)dy
0
Particle in a box
 n ( x)  A sin( kn x)
L
2
|

(
x
)
|
dx  1

0
kL
kL
kn L
1
0 sin ( y)dy  2 0 (1  cos(2 y))dy  2
2
2 kL
A
1
k
A
2
2
A kL A L
0 sin ( y)dy  2k  2
2
2
L