VIT, Vellore
BPHY101P - Engineering Physics Lab Manual
Demonstration of the discrete energy levels and the wave
functions using Schrödinger equation
Tools required:
Excel file with principle quantum number (n) and energy values (E).
Inferences from Theory:
1. When a given particle is confined to a finite space, its momentum (p) and energy (E)
are discrete. The minimum energy or the ground state energy is called “zero-point
energy” which is not zero.
2. In the ground state, the particle has highest probability at the center of the
confinement (box). This idea can be extended to the 3-dimensional confinement also.
3. The number of positions with the maximum probability will increase as the particle
attains higher energy. The number of positions with maximum probability is
proportional to the quantum number “n”.
4. Classical limit is obtained when we consider very high values of quantum number
which will have “n” possible values of maximum probability. In other words, “n”
tends to infinity, there will be infinite positions with maximum probability. This also
means that all the possible positions within the confinement is equally probable. The
corresponding wave function and Energy are as follows.
2
𝜓𝑛 = √𝐿 sin
𝑛𝜋𝑥
𝐿
and 𝐸𝑛 =
𝑛2 ℎ 2
8𝑚𝐿2
where n takes values 1, 2, 3, ….
Observations:
Experiment on quantum confinement: Finding the wavefunctions and probability
1. Calculate the magnitude of energy values for ground state and first 3 excited states
(total 4 energy levels) for a particle confined to a one-dimensional space of 10 nm.
2. Plot the wavefunctions and probability for n = 1, 2 and 3. Use the excel file. Please
note that for simplicity, we have considered m* = 1 in the calculations provided in
the excel file.
3. Use the excel file find out the wavefunction and probability plots and attach the
pictures of both wave function and probability in your record book.