Bandwidth theorem
∆ω∆t ≈ 2π
[8.1]
(Width in frequency domain)⋅(Width in time domain) ≈ 2π Remember : ω = 2πf
so
∆f ∆t ≈ 1
[8.2]
These relations are properties of all waves including particle waves in quantum mechanics.
Bandwidth theorem implies that any wave phenomenon that occurs over a time interval ∆t has to
given by: have a spread of frequencies ∆f given
∆f = 1/ ∆t Hz If ∆t is small then ∆f will be large.
Example: A clap has a much smaller time time width, ∆t, than a cough. Therefore a clap has a much larger frequency spread than a cough. The ear is very sensitive to different different frequencies and so can easily distinguish between these two sound wave packets.
Complicated amplitude distributions: The bandwidth theorem applies even if the frequency domain amplitude distribution is complicated .
Take A(ω) to be Gaussian as an example: 2
A(ω ) = exp(−(ω − ω 0 )2 / 2σ )
Add together N waves with different wavelengths, but with Gaussian amplitude distribution c 10000 Speed the same for all wavelengths
λ low 50
λ max 200
Put in a distribution of amplitudes, make it Gaussianaround ωmean, with a width σ
σ = 90 A( ω )
ω mean = 584.126 exp
(ω . 1
ω mean )
2
2 . 2 σ
A ωn
ωn
A ω n . sin k n. x
φ( x , t) n
ω n. t
φ( 0 , t )
t
-0.1
0.1
The 'width' of the amplitude distribution is σ( i.e dω). From the graph ,pulse width δt ~ 0.07 s, thus
σ . 0.07 = 6.3
i.e ~ 2π
So the bandwidth theorem works!
Bandwidth theorem in real waves In real physical situations the width δω of the amplitude distribution in the frequency domain, and the width ∆t of the wave packets in the time domain, may be very hard to define.
In most cases these quantities will be estimated as being wider than they are. => In general one will find : ∆ω∆t >> 2π You will never find : ∆ω∆t << 2π You can not beat the bandwidth theorem: it is a
fundamental limit (as is the 2nd law of thermodynamics)
More bandwidth theorem We have seen that ∆ω∆t ~ 2π Can also look at spread in k values (k = 2π/λ) ∆k and the spread in space, ∆x, of the wave packet
∆k ∆x ≈ 2π [See tutorial question]
To superpose waves with different amplitudes at different frequencies we have been using the formula: A ω n . sin k n. x
φ( x , t)
ω n. t
n This is a simple case of the Fourier Theorem. The most general form is to write
Φ ( x, t ) = A0 + A( ) sin( kx − t ) + B( ) cos(kx − t ) So the sine and cosine terms each have independent amplitude distributions A(ω) and B(ω)
Adding the whole Fourier Series: add together a large number of waves with both sine and cosine terms c
1000
N
50
δω
All have same speed
λ low ω0
ωN 2
A( ω )
.9
B( ω )
.4
70
λ max
130
width of amplitude distribution
Fixed amplitude distributions
Sum the N waves, each having its own amplitude A0
100 This will turn out to equal the MEAN VALUE of wavepacket φ(x,t)
φ( x , t)
A ω n . sin k n. x
A0 n
ω n. t
B ω n . cos k n. x n
150
Notice how hard it is φ ( 0 , t ) 100
to estimate the width of
A0
this wavepacket in the
50
time domain, also note 0
1
0 t
1
that the mean value of the wavepacket is given
by the first ( constant) term in the Fourier series
ω n.
As before but with random amplitude distributions
A( ω )
rnd( 1 )
B( ω )
rnd( 1 )
Put in some RANDOM amplitude distributions
Frequency domain 1 1 A( ω ) 0.5 B( ω ) 0.5 0
40
60
80
100
0
ωn
40
60
80 100
ωn
Time domain 150
φ ( 0 , t ) 100 A0
50 0
1
0
1
t
Very similar wave packet even with random input amplitudes: ∆t still determined by the input ∆λ or (∆k).
Consequences of the Bandwidth Theorem Width of spectral lines Atoms in a gas discharge lamp emit light over a finite time interval : ∆t ∼ 10−8s Thus the light emitted is not mono-chromatic since there has to be a spread of frequencies:
∆f ~ 1/ ∆t ~ 108 Hz This spread will make the spectra line have a finite width, called the natural line width (see Hecht section 7.10)
Other reasons causing line broadening
Doppler effect: The atoms in a hot gas have a speed
(V ms-1) given by: 1/2 m V2 = 3/2 kT where k is Boltzmann’s constant, T is the temperature in Kelvin, m is the mass of the gas molecule in kg. Waves emitted by atoms moving towards the observer are blue shifted (wavelength decrease) and waves emitted by atoms moving away from the observer are red shifted. This process often dominates over the natural
line width.
Collision Broadening: Collisions between atoms will disturb the energy levels and cut short the emission processes (∆t smaller). Hence ∆f must be larger.
y i
x i
Observe random phase changes as the atom collides with other atoms. Reduces the effective decay lifetime.
Bandwidth Theorem in Quantum mechanics Particles exhibit wave behaviour Momentum p = h/ λ = 2πh / 2πλ = k Where h is Planck’s constant, = h/2π Also energy E = hf = ω As waves these must obey the bandwidth theorem
i.e.
∆k ∆x ≥ 2π
=>
∆k ∆x
≥ h
Using ∆k = ∆p, the spread in momentum =>
∆p∆x ≥ h
This is Heisenberg’s uncertainty principle. Also =>
∆ω∆t ≥ 2π, hence ∆k ∆x ≥ h ∆E∆t ≥ h
In QM this means that for very short time intervals then the strict conservation of energy can be violated
∆E ≤ h/ ∆t
