Understanding the Significance of Wavefunction and Wavefunction Plo...

Greetings to everyone! In my previous blog, we have studied about Schrodinger's wave equation and derived a solution to it by considering a one dimensional system. We were able able to find the value of normalization constant and also successfully found the energy eigenvalue. We almost completed the 1D box, just I thought, I need to give a visual representation of wavefunction plots for 1D box. This way, we will be able to grasp the concepts better. So, today, we shall be discussing about wavefunction plots for 1D box.

Copy of Copy of Stemsocial ChemFam DEC (10).gif

Significance of Wavefunction

In quantum mechanics, the wavefunction is a mathematical function that represents the quantum state of a particle. The square of the wavefunction, denoted as |ψ|², gives the probability density of finding the particle at a particular position. The wavefunction contains information about the particle's position, momentum, and other observable properties.

The wavefunction, often represented by the symbol ψ, evolves over time according to the Schrödinger equation, a fundamental equation in quantum mechanics. By solving the Schrödinger equation for specific systems, scientists can obtain wavefunctions that describe the behavior of particles in those systems.

Ψ actually has no physical significance.
Ψ² gives the probability of finding electron in a certain area.
ΨΨ* dτ gives the probability of finding the electron in a small volume dτ
ΨΨ* is always positive and real.

Important note

When Ψ² value is very high, it means that the probability of finding the electron is very high in those regions. These regions are therefore termed as atomic orbitals. Now, the regions corresponding to Ψ² value equal to 0, are the space where probability of finding the electron is zero i.e., null. These regions are called as node(singular) or nodes(plural).

Wavefunction Plots for the 1D Box:

To gain insights into the behavior of a particle in a 1D box, physicists often rely on graphical representations of the wavefunction. These plots provide a visual understanding of how the probability density changes along the confined space.

Asymmetric 1D Box

For Ground State

It is important to note that in particle in a box, ground state belongs to the state where n=1 and not n=0.

math-20240129 (1).png
For maximum value, sin(πx/l) should be maximum.
i.e., at sin 90°=1= sin(π/2)
So,
math-20240129 (2).png
Therefore, at x=l/2, the value of wavefunction ψ₁(x) will be maximum.

Plot of wavefunction vs length of the box for ground state

For First Excited State

n=2 for first excited state

math-20240129 (8).png

So, for maximum value again, sin(2πx/l) should be maximum.
i.e., at sin 90°=1= sin(π/2)
So,
math-20240129 (3).png
Therefore, at x=l/4, the value of wavefunction ψ₂(x) will be maximum.

Now, for x=l/2,

math-20240129 (9).png

Again for x=3l/4,

math-20240129 (10).png

plot2 (3).png

Plot of wavefunction vs length of the box for first excited state

For Second Excited State

n=3 for second excited state.

math-20240129 (11).png

So, for maximum value again, sin(3πx/l) should be maximum.
i.e., at sin 90°=1= sin(π/2)
So,

math-20240129 (4).png

Therefore, at x=l/6, the value of wavefunction ψ₃(x) will be maximum.

plot3 (1).png

Plot of wavefunction vs length of the box for Second excited state

Probability Plots

|Ψ|² is definitely a positive quantity, so the plots will lie on or above the positive axes only.

Symmetric 1D Box

Always go with the formulas for asymmetric box, if the question does not mention whether the box is asymmetric or symmetric 1D box.

math-20240129 (12).png

math-20240129 (13).png

math-20240129 (14).png

For Ground State

For, ground state, n=1,

math-20240129 (15).png

For maximum value, cos(πx/2l) should be maximum.
i.e., at cos 0°=1= maximum
So,
math-20240129 (5).png

Therefore, at x=0, the value of wavefunction ψ₁(x) will be maximum.

For First Excited State

n=2 for first excited state

math-20240129 (18).png

So, for maximum value again, sin(2πx/2l) should be maximum.
i.e., at sin 90°=1= sin(π/2)
So,
math-20240129 (6).png
Therefore, at x=l/2, the value of wavefunction ψ₂(x) will be maximum

Plot of wavefunction vs length of the box for first excited state

For Second Excited State

n =3 for second excited state

math-20240129 (17).png
So, for maximum value again, cos(3πx/2l) should be maximum.
i.e., at cos 0°=1= maximum
So,
math-20240129 (7).png
Therefore, at x=0, the value of wavefunction ψ₃(x) will be maximum.

plot5 (1).png

Plot of wavefunction vs length of the box for Second excited state

What we learnt?

We studied the significance of wavefunction, the probability of finding an electron in a certain region. We were able to understand what are nodes and what are atomic orbitals.
We studied the wavefunction plots for particle in a asymmetric one dimensional box. We studied the ground state, first excited state and the second excited state and found out the length of the box for which wavefunction will be maximum.
We studied the wavefunction plots for particle in a symmetric one dimensional box. We studied the ground state, first excited state and the second excited state and found out the length of the box for which wavefunction will be maximum.