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FI 3103 Quantum Physics
Alexander A. Iskandar
Physics of Magnetism and Photonics Research Group
Institut Teknologi Bandung
Basic Concepts in Quantum Physics
Probability and Expectation Value
Heisenberg Uncertainty Principle
Wave Function in Momentum Space
Alexander A. Iskandar Basic Concepts in Quantum Physics 2
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Review on Probability Recall the concept of probability density function.
Consider the following example • A manufacturer of insulation randomly selects 20 winter days and
records the daily high temperature
24, 35, 17, 21, 24, 37, 26, 46, 58, 30,
32, 13, 12, 38, 41, 43, 44, 27, 53, 27
• Put in a class table
Alexander A. Iskandar Basic Concepts in Quantum Physics 3
Class Ti Freq. Rel. Freq. (fi)
10 < T 20 15 3 0.15
20 < T 30 25 6 0.30
30 < T 40 35 5 0.25
40 < T 50 45 4 0.20
50 < T 60 55 2 0.10
N = 20 1.00
0
5
10
5 15 25 35 45 55 MoreF
req
ue
nc
y
Histogram: Highest Temperature
Probability Density Function
Review on Probability Consider the following example
• A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature
24, 35, 17, 21, 24, 37, 26, 46, 58, 30,
32, 13, 12, 38, 41, 43, 44, 27, 53, 27
• The average temperature can be calculated using the probability density function as
thus,
Alexander A. Iskandar Basic Concepts in Quantum Physics 4
.)..)(.(
))(.(
functdistprobvaluemid
dataofnumber
frequencyvaluemidaverage
N
fPPT
N
fTT i
iii
ii
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Review on Probability • One other important statistical quantity is the standard deviation,
or,
Alexander A. Iskandar Basic Concepts in Quantum Physics 5
22
22
22
2
22
2
2
2
TTTT
PTPTTPT
PTTTT
N
fTTTT
iiiii
iii
ii
222 TT
Probability Interpretation of Wave Funct. As stated by Born, the modulus of the wave function
of a particle is interpreted as the probability density function associated with the particle
Then the wave function has to satisfy the following
However, one can always normalize the wave function by multiplying it with a constant, .
Hence the condition needed to be satisfied by the wave function is that the initial state of the wave function must be a square integrable function
Alexander A. Iskandar Basic Concepts in Quantum Physics 6
dxtxdxtxP2
,),(
),( tx
),( tx
1,),(2
dxtxdxtxP
),( txN
dxx2
0,
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Probability Interpretation of Wave Funct. Example : Normalize the wave function
To normalize the wave function, we multiply it with a number N, and impose the normalization condition
Thus, the normalized wave function is
Alexander A. Iskandar Basic Concepts in Quantum Physics 7
elsewhere
LxxLxtx
0
0
)(),(
25
52
0
22342
222
303
1
4
2
5
12
,,1
NL
LNdxxLLxxN
dxtxNdxtxN
L
LxxLxL
txNtx 0,)(30
),(),(~5
Expectation Value from Probability In analogy with probability concept, the expectation value of
the particle’s position is
Or in general expectation value of any function f(x) should be calculated as
And uncertainty of the particle’s position measurement is
Alexander A. Iskandar Basic Concepts in Quantum Physics 8
dxtxxtxdxtxxdxtxxPx ,,,),( *2
dxtxxftxxf ,)(,)( *
2
*2*
222
),(),(),(),(
dxtxxtxdxtxxtx
xxx
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Expectation Value from Probability Example : For the normalize the wave
function
Calculate , and hence x.
Alexander A. Iskandar Basic Concepts in Quantum Physics 9
elsewhere
LxxLxtx L
0
0
)(),( 5
30
26
1
5
2
4
1302
30
)(30
,,
0
5432
5
0
23
5
*
LLdxxLxxL
L
dxxLxL
dxtxxtxx
L
L
x 2x
Expectation Value from Probability Example : For the normalize the wave
function
Calculate , and hence x.
Alexander A. Iskandar Basic Concepts in Quantum Physics 10
elsewhere
LxxLxtx L
0
0
)(),( 5
30
7
2
7
1
6
2
5
1302
30
)(30
,,
22
0
6542
5
0
24
5
2*2
LLdxxLxxL
L
dxxLxL
dxtxxtxx
L
L
x 2x
724
1
7
222 LLxxx
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Conservation of Probability If we normalize the wave function at one time, will it stay
normalized? I.e. does
hold for all time? In short, is probability conserved?
Take a time derivative of the probability density function
Use the Schrodinger equation to replace the time derivative and assume that the potential function V(x) is real.
Recall the complex conjugate of the Schrodinger equation
Alexander A. Iskandar Basic Concepts in Quantum Physics 11
1,2
dxtx
t
txtxtx
t
tx
t
tx
t
txP
),(
),(),(),(),(),(
2
),()(),(
2
),(2
22
txxVx
tx
mt
txi
Conservation of Probability Then
Define the probability flux or probability current as
Hence, we get
Alexander A. Iskandar Basic Concepts in Quantum Physics 12
xximx
xxmi
t
txtxtx
t
tx
t
txP
2
2
1
),(),(),(
),(),(
2
2
2
22
xximtxj
2),(
0),(),(
x
txj
t
txP
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Conservation of Probability If we integrate over all space, we get
The last step follows from the fact that for square integrable function j(x,t) vanishes at .
Alexander A. Iskandar Basic Concepts in Quantum Physics 13
0),(
),(
),(),(2
dxx
txj
dxt
txP
dxtxPt
dxtxt
Probability Current
The last relation is similar to the continuity equation found in classical mechanics or electromagnetism, which states that probability is conserved not only globally but also locally.
It means that if the probability of finding particle at a certain point decreases, this probability does not only turns up at another point, but instead it flows to this other region.
Hence the name probability current for
Alexander A. Iskandar Basic Concepts in Quantum Physics 14
0),(),(
x
txj
t
txP
xximtxj
2),(
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Expectation Value of Momentum How do we calculate the expectation value of the momentum
of the particle associated with the wave function (x,t)?
Recall the classical expression for momentum
Take the expectation value of this expression, yields
Note that, the position x does not have a time dependence.
The time dependence of comes from the time dependence of (x,t).
Alexander A. Iskandar Basic Concepts in Quantum Physics 15
p
dt
dxmmvp
dxt
txxtxtxx
t
txm
dxtxxtxdt
dmx
dt
dmp
),(),(),(
),(
),(),(
x
Expectation Value of Momentum Use the Schrodinger equation, to obtain
Note that
Alexander A. Iskandar Basic Concepts in Quantum Physics 16
dxx
txxtxtxx
x
tx
i
dxt
txxtxtxx
t
txmp
2
2
2
2 ),(),(),(
),(
2
),(),(),(
),(
2
2
2
2
xx
xxx
x
xxx
xx
xx
xxx
xxx
x
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Expectation Value of Momentum Thus,
Since the wave function is square integrable function, it means that it vanishes at , hence the first term does not contribute to the evaluation.
Thus, we have
which suggest to associate the momentum to the operator
Alexander A. Iskandar Basic Concepts in Quantum Physics 17
dxxix
xxxi
dxxi
dxx
xxxxi
p
2
222
dx
xip
xip
Expectation Value of Momentum Example : For the normalize the wave
function
Calculate , and hence p.
Alexander A. Iskandar Basic Concepts in Quantum Physics 18
elsewhere
LxxLxtx L
0
0
)(),( 5
30
p 2p
03230
)2)((30
),(),(
0
223
5
0
5
L
L
dxxLLxxiL
dxxLxLxiL
dxtxxi
txp
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Expectation Value of Momentum Example : For the normalize the wave
function
Calculate , and hence p.
Alexander A. Iskandar Basic Concepts in Quantum Physics 19
elsewhere
LxxLxtx L
0
0
)(),( 5
30
p 2p
2
2
0
2
5
2
0
5
2
2
222
1022
30
)2)((30
),(),(
LdxLxx
L
dxxLxL
dxtxx
txp
L
L
Expectation Value of Momentum Example : For the normalize the wave
function
Calculate , and hence p.
Alexander A. Iskandar Basic Concepts in Quantum Physics 20
elsewhere
LxxLxtx L
0
0
)(),( 5
30
p 2p
Lppp
10
22
72
Lx Recall the previous results of , then
Which is consistent with .
6.072
10
L
Lxp
2 xp
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Expectation Value of Momentum The expectation value of position x is a real value, as can be
easily seen from the definition,
However, because of the form of momentum operator that involves the imaginary number as well as a differentiation,
will the expectation value of the momentum be a real value?
In fact, it can be proved that the momentum expectation value is always a real number.
Alexander A. Iskandar Basic Concepts in Quantum Physics 21
dxtxxdxtxxPx2
,),(
dxtx
xitxp ),(),(
Expectation Value of Momentum
In the last step, the square integrability property of the wave function, (x,t), has been used, where it states the the wave function is a localized function so that (x,t) 0 as x .
Alexander A. Iskandar Basic Concepts in Quantum Physics 22
0),(),(),(),(
),(),(
),(),(*
),(),(
),(),(*
),(),(*),(),(*
**
*
*
*
txtxi
dxtxtxxi
dxx
txtx
x
txtx
i
dxx
tx
itx
x
tx
itx
dxtxxi
txdxtxxi
txpp
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Heisenberg Uncertainty Principle The result of the last example on product of the expectation
values of x and px, it was found that
This relation is called the Heisenberg Uncertainty principle.
It states one of the fundamental concept of quantum world, that is measurement of position and linear momentun cannot be done simultaneously with the highest accuracy.
If measurement of position is done very accurately, x = 0, then the value of the linear momentum is not known, since according to Heisenberg uncertainty principle yields px , and vice versa.
Position and momentum are said to complementary variables.
Alexander A. Iskandar Basic Concepts in Quantum Physics 23
2
xpx
Heisenberg Uncertainty Principle One observation can be used to
see this principle.
From wave optics, the spread of the diffraction pattern as
Considering the light as photons, at the slit, the position of the
Alexander A. Iskandar Basic Concepts in Quantum Physics 24
2
2
2xp
2xpa
photon is within the measurement’s uncertainty of x a, but the momentum’s uncertainty has spread.
Thus
where .
hxppapap
pxxx
x
x
2 kpx
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Heisenberg Uncertainty Principle Note that the Heisenberg uncertainty principle between the
position and momentum holds for all direction (x, y and z).
One other Heisenberg uncertainty relation is the uncertainty between energy and time
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2
tE
E
t
21
E
ground state
excited state
E
ground state
excited state
Wave Function in Momentum Space Recall that the spectral distribution function of the wave
packet is none other than the Inverse Fourier transform of the wave function at t = 0,
Calculate the following
Alexander A. Iskandar Basic Concepts in Quantum Physics 26
dxexppx
i
)0,(
2
1)(
1)()(
)(2
1)(
)(2
1)()()(
dxxx
dxdpepx
dpdxexpdppp
pxi
pxi
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Wave Function in Momentum Space
Fourier transform of a normalized wave function is normalized.
Next, consider the following
Which is a statement that the momentum expectation value can also be calculated from (p) using the momentum operator p itself.
Alexander A. Iskandar Basic Concepts in Quantum Physics 27
1)()()()(
dxxxdppp
dppppdpdxexpp
dxdpepdx
d
ix
dxxxi
xp
pxi
pxi
)()()(2
1)(
)(2
1)(
)()(
Wave Function in Momentum Space
This last statement is similar to
Thus if (x,t) is the wave function in spatial domain, (p) should be interpreted as the wave function in momentum space, with the probability density function of finding a particle with momentum p is given by |(p)|2.
In this momentum space, the position operator is given by
Hence, the expectation value of position is
Alexander A. Iskandar Basic Concepts in Quantum Physics 28
dppppp )()(
dxtxxtxx ,,*
dpp
pipx )()(
pi
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Wave Function in Momentum Space Example 2.4
Consider a particle whose normalized wave function is
Alexander A. Iskandar Basic Concepts in Quantum Physics 29
elsewhere
Lxxetx
x 0
0
2),(
x 2x
p 2p
a. For what value of x does P(x) = |(x)|2 peak?
b. Calculate and .
c. What is the probability that the particle is found between x = 0 and x = 1/?
d. Calculate (p) and use this to calculate and .
x 1/
(x)
Summary Physical quantity (an observable) is represented by an
operator.
Measurement of observable is evaluated as calculating expectation value
There are two ways to calculate the expectation value, in spatial space or in momentum space.
Alexander A. Iskandar Basic Concepts in Quantum Physics 30
dxtxtxquantityphys ),(ˆ
),(. O
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Summary
Alexander A. Iskandar Basic Concepts in Quantum Physics 31
Spatial space Momentum
space
Wave function
Position
Momentum
),( tx )( p
xi
pi
x
p
dpeptxEtpx
i)(
)(2
1),(
dxexppx
i
)0,(
2
1)(