The simplest Schrödinger equation, describing a free particle confined to move in a one-dimensional — Physical Chemistry — Electrochemistry Chemistry Question
Schrödinger equation
The simplest Schrödinger equation, describing a free particle confined to move in a one-dimensional ‘rigid box’ brings out a most basic fact: quantization arises due to boundary conditions on the wave function.
An electron of mass m is confined to move in a line along the x-axis from x = 0 to x = L. Between the two ends it experiences no force.
Write down the (time-independent) Schrödinger equation for the wave function of an electron.
Model Answer
One-dimensional Schrödinger equation for a free particle of mass m:
d2ψ/dx2 + (8π2m/h2)Eψ = 0
where E stands for the energy of the particle and ψ its wave function.
Which of the following are possible wave functions of an electron in one-dimensional rigid box:
e^(-kx) , cos(nπx/L) , sin(kx) , sin(nπx/L)
where k is any real number and n is a positive integer ?
Model Answer
The boundary conditions are:
ψ(0) = ψ(L) = 0
Only ψn(x) = sin(nπx/L) satisfies the required boundary conditions.
Other functions are not possible wave functions of the electron in a one-dimensional rigid box.
For the acceptable wave functions of the electron in (ii) above, show that the energies are given by
En = n^2 h^2 / 8mL^2
Model Answer
d2/dx2 (sin nπx/L) = - (n2π2/L2) sin (nπx/L)
En = n2h2 / 8mL2
Plot schematically the wave function of the electron in the ground and the first two excited states. What is the number of nodes (in the region between x = 0 to L) of the wave function with energy En?
Model Answer
Number of nodes in ψn = n – 1, apart from the nodes at the end points.
Ground state (n = 1)
ψ1(x) = sin(πx/L)
First excited state (n = 2)
ψ2(x) = sin(2πx/L)
Second excited state (n = 3)
ψ3(x) = sin(3πx/L)
Normalize the ground state wave function of the electron.
(The integral of the square of the modulus of a normalized wave function over all space is unity.)
Model Answer
∫ |ψ(x)|² dx = 1
N² ∫ ² dx = N²(L/2) = 1
N = √(2/L)
ψ1(x) = √(2/L) sin(πx/L)
An interesting example of this one-dimensional model in chemistry is the motion of an electron in a conjugated system of single and double bonds. The molecule 1,3-butadiene has four electrons assumed to move freely in a line consisting of three carbon-carbon bonds, each of approximately the same length (1.4×10–10 m), with an additional length of 1.4×10–10 m at each end.
Using the aufbau principle, determine a scheme to fill the electrons in the available energy levels. Calculate the lowest excitation energy of the system.
Model Answer
L = 5 × 1.4×10^-10 m = 7.0×10^-10 m
The first three energy levels are:
E1 = h^2 / 8mL^2 = 1.22×10^-19 J
E2 = 4 E1 = 4.88×10^-19 J
E3 = 9 E1 = 10.98×10^-19 J
In the ground state, the four electrons will occupy the levels E1 and E2, each with two electrons.
Ground state Lowest excited state
The lowest excitation energy
E3 – E2 = 6.10×10^-19 J
Boundary conditions on wave functions result in quantization of not only energy but also other physical quantities, such as angular momentum. The wave function corresponding to the value hλ / 2π for the z-component of angular momentum (Lz) is:
ψ(φ) = e^(iλφ) ,
where φ is the (azimuthal) angle in the x-y plane measured relative to the x-axis.
Use the condition that this function is single valued at every point in space and show that this implies that λ is quantized. Give the quantized values of angular momentum projection along the z-axis.
Model Answer
The condition that ψ(φ) is single valued demands that
ψ(φ) = ψ(φ + 2π)
e^(iλφ) = e^(iλ(φ + 2π))
e^(i2πλ) = 1
i.e. λ = m, where m = 0, ±1, ±2, ±3,…….
This shows that angular momentum projection (Lz) cannot be an arbitrary real number but can have only discrete values: m, where m is a positive or negative integer (including zero).