Applications in Quantum Mechanics

In quantum mechanics it is useful in

1. Calculating bound state energies (Whenever
the particle cannot move to infinity)
2. Transmission probability through potential
barriers.
These are given in next slides.
• If V (x) is not constant, but varies slow in comparison
with the wavelength λ in a way that it is essentially
constant over many λ, then the wave function is
practically sinusoidal, but wavelength and amplitude
slowly change with x. This is the inspiration behind
WKB approximation. In effect, it identifies two
different levels of x-dependence :- rapid oscillations,
modulated by gradual variation in amplitude and
wavelength.
 The WKB approximation

V(x)

Turning points

E
 𝑝2
𝐴′′+ 2𝑖𝐴′𝜙 ′+ 𝑖𝐴𝜙 ′′− 𝐴 𝜙 ′2 = − 2𝐴
ℏ
Solving for real and imaginary parts we get,
2
𝑝
𝐴′′− 𝐴 𝜙 ′2 = − 2 𝐴
ℏ
2
𝑝
𝐴′′= 𝐴 ( ∅ ′ )2− 2
ℏ
𝐴′ 𝑝 2 ′
= (∅′ )2− 2
𝐴 ℏ
The above equation cannot be solved in general, so we use WKB
approximation: we assume amplitude A varies slowly, so that the
′
𝐴′ 2 𝑝2
A’’ term is negligible. We assume that 𝐴 << [( ∅ ′
) −
ℏ2
].
Therefore, we drop that part and we get
1
𝜙 𝑥 = ± 𝑝 𝑥 𝑑𝑥
ℏ
 Example 1

Potential Square well with a

Bumpy Surface
 Example 2

Tunneling
Note: the following example is alpha decay in more
details to know about the energy calculations
Example:
Chapter 11: Quantum Dynamics

Thus far, our potentials V(r) have been time independent.

Now we will use V(r,t).

These type of problems are very difficult so we will

work in the context of perturbation theory. The most
dominant portion of H will be time independent, while
only the perturbation will be time dependent.

By this procedure we will study the emission and

absorption of radiation (photons) by an atom.
Thus far, we have casually spoken about transitions from
high energy states to low. And you have learned about this
since probably High School. But mathematically, if we are in
an excited state, THUS FAR we stay there forever.
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Then, how do we describe the transitions between levels,
such as when an electron decays from a state, say, n=3 l=1
to the ground state n=1 l=0, by emitting a photon?

At the minimum, we have to “shake” a little bit the

electron at n=3 l=1 to allow for the transition to occur.
How do we shake the electron? By placing the electron in
an external electromagnetic field, which often is time
dependent. Then, “quantum jumps” – that are almost
instantaneous – can occur.

11.1: Two-level systems

To study emission and absorption we need at least two

states. Consider two eigenstates of H0:

2
We will assume these two states are the only ones that
really matter. Thus, any arbitrary state can be a linear
combination of “a” and “b” at time t=0:

Again, you can imagine these two states as for example

the 1s and 2p states of the hydrogen atom. Or they can be
spinors up and down, describing a spin in a magnetic field.

If there is NO perturbation, you learned in the fall

QM P411
I that
the time dependence is easy:

So far nothing new …

3
 Now we will introduce a that
depends on time, such as an small external field.
The two original states are still a complete basis. But the
time dependence is not so easy. We will consider the new
time dependence by making the coefficients, that before
were fixed numbers, time dependent...

We know a, b, Ea, and Eb. Thus, the challenge is

to find the coefficients as a function of time.

For example if at time t=0, ca(0)=1 and cb(0)=0, then the electron is
at “a” initially. If at a later time T the coefficients are ca(T)=0 and
cb(T)=1 a transition occurred from a to b.
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 How do we find ca(t) and cb(t)? Requiring that the time
DEPENDENT Sch Eq be satisfied.

Recalling
we simply plug this state into the Sch. Eq. above:
Time independent Hamiltonian Perturbation is time dependent.
like hydrogen atom. It may be ~ sin( t) for example.

Time dependent
coefficients. The dot means
dca(t)/dt or
dcb(t)/dt .

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Rewriting the formula we can notice some cancellations.

Then, the long original equation simplifies to:

As often done, we will now

exploit the orthogonality

6
 Consider the inner product with < a|
and then the inner product with < b| :

Then we obtain two equations:

Reorder and use the following

compact notation:

Note sign
difference.
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 This system of two coupled equations for the coefficients ca(t)
and cb(t) is totally equivalent to solving the time dependent
Sch Eq . If you
wish to include more states, then more equations are
generated, as many as coefficients i.e. as many as states.
Moreover, as you will see, often the diagonal matrix elements
aa and bb are 0. E.g. an electric field arises from a potential
that is “odd”: .Then a “diagonal” matrix element
involving odd or even functions will cancel.

Then, often in practice the system of

equations simplifies further to:

b
where with
a 8
 11.1.2: Time Dependent Perturbation Theory

It is only now that we will assume is “small”. We

will use an iterative process …

.
b

Start with:
a
If , then all the matrix elements

The particle remains at state “a”.

Now insert the 0th-order values

for the coefficients into the
right of the pair of differential
Eqs. of previous page:

For “a” nothing changes:

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 For “b”, we find a nontrivial result:

To obtain something nontrivial for the “a”

We could continue the process, but this is sufficient. Please

read in book page 406-407 the discussion about the

to 1 is not respected.
are keeping.
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 11.1.3: Sinusoidal Perturbation

Consider, very common, perturbations

where the space and time component
are separated in factors:

Then in this case, we find exactly at order 1 for the b coefficient:

Another simplification. Work near resonance and drop the first term.

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The transition probability for the transition from “a” to “b” is a
sinusoidal function of time that can be large near resonance, even
with a small perturbation Vba.

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 .
b
.
b

.
b

a
a

.
b

a
a

The amplitude, meaning how likely is the transition from “a” to “b”,
is regulated by both the perturbation strength hidden in Vba and
also by how close to resonance we are. If the amplitude exceeds 1,
then the formula is too crude and needs to be improved …

Moreover, the probability is sinusoidal. The particle can absorb

energy and go up, or release energy and come down.

At times where n=1,2,3, … the electron is back

in the lower state “a” with 100% chance. Thus, often it is better to
turn off the external field, after a time sufficient to excite the
electron, if you wish to keep the electron in the upper state.

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Plotting results at a fixed time, as a function of frequency, make
more clear that near resonance the probability is maximized.

Near resonance, ~[ ]2

At resonance, amplitude grows like t2 so

eventually the transition will surely occur.
1
 11.2: Emission and
absorption of radiation

11.2.1:
Electromagnetic waves

Polarized light
along z axis

If atom is much smaller than wavelength, then

the electric field is ~ uniform inside atom
2
 Visible light is ~5000 Å
while an atom is ~ 1 Å
≠
E=- H’(r)/q

We will see that diagonal matrix

elements of H’ vanish by symmetry for
atoms such as hydrogen. Thus the
matrix elements that matter are (a≠b):

where the electric dipole moment →

Relating with previous generic

formulas thus require replacement →

3
Suppose we repeat the same calculation b
as in the previous lecture but with the
electron first at b. Result is the same
.
just switching a < -- > b.
a

As expressed before, even though you are originally at “b”, by

merely being immersed in a radiation field (i.e. a lot of
photons) the electron can decay to a lower energy.

This is called stimulated emission. Absorption

The electron in “b” is “unstable”, (previous page)
and any perturbation (like a
shower of photons) may drop it to Stimulated
“a”. Actually the probability a->b emission
and b->a is the same.
4
Because of stimulated emission amplification can occur. Almost
instantly a huge number of photons in phase can be produced:

You need three states in practice:

This is the basis of

Light Amplification by
Stimulated Emission of
Radiation (LASER).

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 So far, in the presence of the same external field of the right
frequency, (1) we can either move an electron from a lower level to
an upper level (absorption) or (2) we can lower an electron from an
upper level to a lower level (stimulated emission).

Remarkably, there is a third possibility: spontaneous emission.

This is when an electron is in a high energy level, there is NO
external field, and yet the electron decays to a lower level
spontaneously.
This contradicts what I said at the start of Ch.11: that
an electron in an upper level will remain there forever
without any disturbance shacking the electron. But in
QUANTUM electrodynamics, like in a harmonic
oscillator, there is always some oscillation present.
The zero-point energy.
Amazingly even in the best laboratory conditions there is always
some radiation present even at zero temperature, similarly as in
ANY quantum mechanics problem: e.g. the particle never rests
at the bottom of the infinite square well because the ground
state energy is not 0 but larger than 0.
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 11.3.3 Selection rules

Very often the matrix elements

that appear in the rate are zero by symmetry.

Consider the hydrogen

atom. In this case:

Commutators discussed time ago (Ch. 4)

allow us to arrive to the first rule:

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From many other commutators learned time ago,
we can derive the second rule:

Although for us the external field is not quantum,

intuitively the results are in agreement with
“photons” emitted or absorbed, because photons
have spin s=1 (bosons) and projections ms = -1,0,1.
Conservation of angular momentum leads to these
selections rules: whatever happens to the electron
in the atom, must be compensated by the photon
with regards to energy and angular momentum.

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 Sometimes you
can only decay
to one state

Spontaneous
Sometimes excited emission (there is
always “some”
? states, like |200>, radiation in Q.E.D.)
cannot decay.

The states that cannot decay are “metastable”

with long lifetimes. They eventually decay from
atomic collisions or emitting two photons (much
lower probability).
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