Introduction to Quantum Mechanics

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Quantum mechanics is the mathematics of atomic and molecular structure. We seek to determine allowed quantum states and their energies.

A Short History

A definite understanding that atoms are composed of separate particles (nuclei and electrons) did not really occur till toward the end of the 19th century. The development of quantum mechanics is strongly tied to the evolving understanding of the nature of matter over many centuries.

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• In 460 BC, atom was first proposed by Democritus.
• In 1897, Thompson discovered electrons.
• 1911-1919, Rutherford’s experiments on scattering showed electrons and protons.
• In 1913, Bohr came up with rules for atomic behavior.
• All of this ultimately led to the concept of the wave function, or wave behavior of matter.
• Wave function then led to quantum mechanics.

So quantum mechanics itself arose out of a number of important experimental and conceptual advances.

1. Waves act like particles. In the late 19th century, Maxwell’s equations was a wave theory. Light was considered to be composed of electromagnetic waves. Any experimental evidence that contrasted with that was somewhat revolutionary.
   • 1900. Planck theory of black body radiation. Planck actually assume that light was being emitted in discrete packets of energy, and that the energy was proportional to a constant called h, and the frequency of light at that particular wavelength. ɛn = n h v
   • 1904. Photoelectric effect. Photon is the term Einstein coined for packets of radiative energy. He could explain experimental observations, assuming that E was equal to h, Planck’s constant times the frequency of the light. ɛ = h v
   • 1923. Compton effect. Photons scatter like particles with momentum p = h / λ
2. Particles act like waves.
   • 1924. De Broglie postulated that particles could act like waves. The wavelength would be equal to Planck’s constant over the momentum of the particle. λ = h / p
   • 1927. Davidson-Germer experiment confirmed that electrons diffracted like waves.
3. Wave particle duality
   • 1924. Schrodinger came up with a way to describe the wave function mathematically.
   • 1927. Heisenberg suggested how to deal with the duality, and this led to the famous uncertainty principle. Describe a particle type behavior with a wave function, which is a function made up of oscillating terms. The wave can travel. It has a contained location, and it moves. The Heisenberg uncertainty principle said that if you try to identify the location of the wave precisely, then there’s some uncertainty in the momentum, and vice versa. If you try to identify the momentum precisely, there’s some uncertainty in location.

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Structure of Matter

Matter is made up of atoms and that atoms can combine to form molecules. Further atoms are composed of a nucleus surrounded by electrons which exist in orbitals called shells. We use the term structure to describe the allowed configuration of an atom or a molecule. It also required experimental determination of a number of behaviors that were unexplained at the time, which is often the case with scientific advancement.

How do atoms and molecules store energy? Molecule can store energy by:

Translation	moving back and forth, up and down, in and out, three coordinate directions. That’s kinetic energy.
Rotation	when it has a moment of inertia about any axis, but only 3 dimensions independently.
Vibration	a diatomic molecule has different vibrational energy, i.e. kinetic and potential energy associated with vibration, but they come in discrete increments.
Electronic motion	moving around the molecule.

In thermodynamics, we’re mainly interested in the kinetic and potential energy of the particles.

We know from quantum mechanics that isolated particles (atoms and molecules) generally exist in particular quantum states, which is an allowed configuration that determines spatial and dynamic behavior of the nuclei and electrons.

There are many situations in which atoms and molecules are undergoing dynamic conditions that cause a change in quantum states, but if we leave them alone long enough, then they will tend to relax to one of the fixed quantum states or eigen states of the system.

Those allowed quantum states can be described by quantum numbers and we need to know the properties of those states. By statistics we know how the large numbers of particles behave in equilibrium.

Energy Level Diagram

We typically describe the structure using energy level diagrams. There are 2 examples: one for an atom, the other one is for a diatomic molecule.

In atomic structure, for historic reasons, zero energy is considered to be when the outermost electron is far enough away from the nucleus, so that they are not interacting. However, as the electrons come closer, there is an actually an attractive force.

So if you think of it as an ion, an electron coming back together, that’s an exothermic reaction, releases energy. As it drops down in the allowed quantum levels, we get more and more energy. For instance for the hydrogen atom, the energy of the lowest state, with comparison to the fully ionized state, is -109678 cm-1. The units of inverse centimeter are commonly used to describe energies in the chemistry and physics world.

In a diatomic molecule structure, because electrons are very light compared to nuclei, that the dynamic behavior of the electrons tends to relax to a steady-state condition much more rapidly than the motion of vibration or rotation. So we can characterize the effect of the electrons of all the electrostatic forces in terms of a potential energy, and it looks something like the line of the Morse potential.

The nuclei are far apart, there’s little interaction if they’re far enough apart. But as you bring them together, then the electrons rearrange themselves in a way that results in a net attractive forces (Van der Waals forces). and they tend to want to pull the nuclei together.

As they get closer to closer together, that attractive force comes into competition with the repulsive force of the two nuclei which have the same charge. Likes oppose, opposites attract. So the nuclei are going to oppose each other. So there’s a balance of the force between them and therefore in the potential energy.

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The Postulates

Each system (an atom or a molecule) can be characterized by a wave function which is complex and contains all of the information that is known about the system. r^→ is position, t is time.

Ψ(r→, t)

The wave function is designed or defined in such a way that the probability p of finding the system in the volume element dV is below, where start * means complex conjugate.

p(r→, t) dV = Ψ*(r→, t) Ψ(r→, t) dV = |Ψ(r→, t)|2 dV

Since we expect to find the system somewhere, size of Ψ or p must be normalized, in other words, the wave function is normalized.

∫ Ψ*(r→, t) Ψ(r→, t) dV = 1

With every dynamical variable, there are associated operators, where ħ is Planck’s constant over 2π, and ∇→ = d/dx + d/dy + d/dz.

Dynamical variables	Operatopors
position r→	r→op = r→
momentum p→ = mv→	p→op = -i ħ ∇→
energy ε	εop = i ħ ∂/∂t
some physical variable B(r→, t)	Bop = B(r→, -i ħ ∇→)

The expectation value of any physical observable is below, where the wave function is determined by using Schrodinger’s wave equation.

⟨B⟩ = ∫ Ψ*(r→, t) Bop Ψ(r→, t) dV

Schrodinger Wave Equation

We can obtain the wave function by solving Schrodinger’s wave equation. Schrodinger obtained the equation by starting with the equation for the total energy of a particle and applying the operators for each term. So basically, the wave equation is a conservation of energy equation. The energy of a particle is the sum of its kinetic and potential energies.

Kinetic energy	K.E. = 1/2 m v2 = 1/2 p2 / m
Potential energy	The form depends on the nature of the particle. We can call it V(r→, t) as it may be a function of space and time.

In classical mechanics, the sum of the kinetic energy and potential energy is called the Hamiltonian:

H = ɛ = p2 / 2m + V(r→, t)

So if we substitute operators in this expression, we end up with is the equation below. This is basically a statement of conservation of energy.

i ħ ∂Ψ(r→, t)/∂t = (-ħ2 / 2m) ∇2 Ψ(r→, t) + V(r→, t) Ψ(r→, t)

Schrodinger needed a linear equation to describe the propagation of wave packets and he wanted the equation to be symmetric with classical conservation of energy. So it’s like conservation of energy where we’ve replaced the classical terms with the operators. The equation describes the time dependent transient motion of any collection of quantum particles subject to body forces including internal motion collision, chemical reactions etc.

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The Stationary Wave Equation

The stationary wave equation comes about when we assume that the potential energy is only a function of spatial position and not time, i.e. V(r→, t) = V(r→), thus we can write below and seek solutions subject to initial and boundary conditions.

i ħ ∂Ψ(r→, t)/∂t = (-ħ2 / 2m) ∇2 Ψ(r→, t) + V(r→) Ψ(r→, t)

Since V is a function of space only, we can use separation of variables Ψ(r→, t) = φ(t) ψ(r→). Substituting this into the wave equation and dividing by φ(t) ψ(r→), we have:

i ħ ∂[ φ(t) ψ(r→) ]/∂t = (-ħ2 / 2m) ∇2 [ φ(t) ψ(r→) ] + V(r→) [ φ(t) ψ(r→) ]
⟹ 1/φ(t) i ħ ∂φ(t)/∂t = 1/ψ(r→) [(-ħ2 / 2m) ∇2 ψ(r→) + V(r→) ψ(r→) ]
⟹ C = C

We get the equation where the left hand side is a function of time and the right hand side is a function only of space. Since the two sides are functions of different variables, they have to be equal to within a constant.

Solving the time dependent part (the left side):

1/φ(t) i ħ ∂φ(t)/∂t = C
⟹ i ∂φ(t)/∂t = C φ(t) / ħ
⟹ φ(t) = exp(-i C t / ħ)

So we can write the total wave function Ψ(r→, t) = ψ(r→) exp(-i C t / ħ). Let us use this to calculate the expectation value of the energy:

⟨ε⟩ = ∫V Ψ* (i ħ ∂/∂t) Ψ dV
= ∫V ψ* exp(i C t / ħ) (i ħ ∂/∂t) ψ exp(-i C t / ħ) dV
= ∫V ψ* ψ exp(i C t / ħ) (i ħ ∂/∂t) exp(-i C t / ħ) dV
= ∫V ψ* ψ exp(i C t / ħ) (i ħ) (-i C / ħ) exp(-i C t / ħ) dV
= ∫V ψ* ψ exp(i C t / ħ) C exp(-i C t / ħ) dV
= C

The separation constant C turns out to be equal to the energy of the system.

We are left with the so-called stationary wave equation, which is a second order partial differential equation. It is subject to appropriate boundary conditions and knowledge of potential energy function V(r→). This is an eigen equation and leads to solutions only for discrete values of the energy.

1/ψ(r→) [(-ħ2 / 2m) ∇2 ψ(r→) + V(r→) ψ(r→) ] = C = ε
⟹ (-ħ2 / 2m) ∇2 ψ(r→) + V(r→) ψ(r→) = ε ψ(r→)

Two Particle Systems

For two particles the total energy is:

ε = p12 / 2m1 + p22 / 2m2 + V(r→)

If we transformed the wave equation into center of mass coordinates, we end up with an equation for the external motion and a separate equation for the internal motion. The energy of two particle system can therefore be divided into:

1. the external energy – the kinetic energy associated with the whole system translating with respect to its center of mass.
2. the internal energy – associated with internal motion such as rotation, vibration, electronic motion, and so on.

External motion	(ħ2 / 2mtot) ∇2 ψext + εext ψext = 0
Internal motion	(ħ2 / 2μ) ∇2 ψint + (εint - V(r→)) ψint = 0

where ε = εext + εint, mtot = m1 + m2, and reduced mass μ = (m1 m2) / (m1 + m2). Note that we’ve assumed that no potential acts on the particle as a whole, that is its translation is not affected by any potential field.

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