Applying Quantum
Mechanics
Lecture 23
Solution of the Shrödinger equation
for a well of finite depth
Recap from the previous lectures
Solving the Schrödinger equation
• The Schrödinger equation
ℏ 𝑑 𝜓
𝑉 𝑥 𝜓 𝑥
𝐸𝜓 𝑥
2𝑚 𝑑𝑥
is solved with appropriate boundary conditions on the wavefunction 𝜓 𝑥 .
• Boundary conditions require (i) the wavefunction to vanish at large distances away
from the system and (ii) continuity of the wavefunction and its first derivative at all
places (for finite potentials).
• Boundary conditions can be satisfied only for certain values of the energy, known as
the eigenvalues. These are the quantized energy levels of the system.
• The corresponding wavefunctions are the eigenfunctions of the system. Square of
their absolute value at a given position gives the probability density of finding the
particle at that point.
A couple of questions that arise and answers to them
Question: While calculating the expectation value 𝑥
𝑥 𝜓 𝑥 𝑑𝑥 what does 𝑥
represent? Is it the position of the particle?
No, 𝑥 is not the position od the particle as talked about in classical physics where a
particle can be assigned a position at a given time. Here in quantum picture, 𝑥 is the
value of a possible position a particle can take with probability 𝜓 𝑥 𝑑𝑥 of being in
that neighbourhood. So it is not a function of time since all values are possible at any
given time. On the other hand, the expectation value 𝑥 can be a function of time.
Question: When we apply the operator 𝑝 for momentum on the wavefunction, do we
get the momentum of the particle?
The answer is no. In general a wavefunction is a combination of many different
planewaves with different amplitudes. So it does not have a unique momentum.
However the expectation value of the momentum operator gives the average value of
the momentum for that state, just like the position of a particle in a quantum system is
not well defined but its average value can be calculated by employing the
wavefunction.
Energy‐time uncertainty relation
• A question that has been asked is about the energy‐time uncertainty
ℏ
and what it means.
relation Δ𝐸Δ𝑡
• First, it is pointed out that in nonrelativistic physics, time is not a
variable associated with a particle but a parameter. Thus the energy‐
time uncertainly relation is not on the same footing as the position‐
momentum uncertainty relation.
• What the relation means is that if in an experiment is carried out within
time Δ𝑡, or an event exists for time Δ𝑡, the corresponding energy will
ℏ
.
have an uncertainty of Δ𝐸
Contd........
Example of 𝜋 meson inside a nucleus was already solved in the previous
lecture, where we estimated the mass of this particle assuming that it exists for
some time 𝜏
where 𝑟 is the radius of the nucleus and 𝑐 is the speed of light.
Because of the time uncertainty of 𝜏, the energy of the system could be
uncertain by an amount ΔE ℏ𝑐⁄𝑟, which is then equated to the extra energy
𝑚 𝑐 needed for the creation of 𝜋 meson with its mass being 𝑚 .
In the next slide, we show the spectrum of Xenon where the intensity of light
emitted by xenon atoms is plotted against the frequency. One sees that for each
emitted line the frequency is not just one number but there is a range of
frequency which is Δ𝜈~ , where 𝜏 is the life‐time of the excited‐state.
Emission spectrum of Xenon
We now solve the Schrödinger
equation for two potentials of
finite depth (or height)