Comparing the Quantum Models
In quantum mechanics, a quantum system can be described by its wave function. The Schrödinger equation is a differential equation that when solved provides the wave function. Solving the Schrödinger equation will help us predict an atom’s, molecule’s, or a subatomic particle’s features including its position, momentum, and energy. We begin with the time-independent Schrödinger equation which allows us to find the stationary states and energy levels. After we acquire this, we can find the position, momentum, and energy of a particle. The time-independent Schrödinger equation: Eψ=Ĥψ involves the Hamiltonian which is denoted by Ĥ. The time-dependent Schrodinger equation allows us to find position, momentum and energy of a particle with respect to time.
The Hamiltonian is the operator* which describes the total energy of the system (kinetic and potential). As a result, each model of a quantum system will have a different Hamiltonian because the potential energy will change with the differing quantum models. This changes the Schrödinger equation as well. As a result, we must compare each of the quantum models, the Hamiltonian, stationary states, and energy levels.
*(To find out more about operators and eigenfunctions and eigenvalues, click on this link, https://260h.pbworks.com/w/page/58805807/The%20Hamiltonian. )
Free Particle
Basic Overview: A quantum particle is said to be free if the expectation value (mean) of its momentum is constant. This particle has no potential energy. Imagine a single particle in a vacuum; it cannot interact with anything. This is more of a theoretical model, since particles like this do not actually exist.
<p>=constant and d/dt<p>=0
Free Particle Hamiltonian: The total energy only dependent on the kinetic energy, since the particle is not bound by any potential energy. When Schrodinger equation is solved, there are no energy levels, no quantization.
- Ĥ=-(ħ2/2m)( ∂2/∂x2)
Eψ(x)=Ĥψ(x)
Particle in a Box
Basic Overview: This theory describes a particle surrounded by four walls that is free to move about anywhere within the four walls. This model is used to describe the differences between classical and quantum models, and illustrates the importance of quantum mechanics on the smaller scales. Simplify this by thinking of the walls as having infinite potential energy, and the inside of the box as having potential energy equal to zero. As L increases, the energy levels get closer to eachother, forming a continuum. As a result, the energy levels are not perceived as quantized energy states and begins to resemble classical mechanics.
- Classical conditions – Imagine the ball is trapped in a large box. The interior of the box is called the “well”. The ball can move any speed and is equally likely to be found at any position within the well as long as the well stays within the scale of classical mechanics.
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Quantum conditions – As opposed to the particle in a classical box that has equal probability to be found anywhere within the box, it is more likely to found in certain places in the quantum (as the well gets smaller, along the lines of a few nanometers) box than others, as well as never occupying some spaces known as, spatial nodes, where the wavefunction equals 0 and therefore the probability that the particles is in that position within the well is also equal to zero.
To learn more about particle in a box and see it in action, visit http://phet.colorado.edu/en/simulation/bound-states.
Particle in a Box Hamiltonian:The potential energy is removed from the Hamiltonian when it is inside the box. By solving the Schrodinger equation with an infinite potential energy, the wave function is equal to zero. Outside of the boundaries ψ(x)=0, the wave function doesn't exist, since the potential energy goes to infinity.
Energy levels increase by n2, so higher energy levels are further apart than lower energy levels.
- Eψ(x)=Ĥψ(x)=-(h2/8π2m)( ∂2/∂x2)ψ(x)
V(x)=0 if a between L=0 an L=a
V(x)=infinity when L smaller than 0 and L bigger than a
Harmonic Oscillator Model
The 1D Harmonic Oscillator Model in Quantum Mechanics describes the wave function of a particle according to Hooke’s Law that states that F=-kx, k being the spring constant.
According to Hooke’s Law the potential energy stored in a particle equals to:
U=1/2*(kx^2)
While the potential energy of the particle changes within the model, the kinetic energy of the former remains the same, as it does not depend on the interaction with other particles in the environment.
The following Schrӧdinger equation is obtained:
Eψ(x)=Ĥψ(x)=[-(h^2/8π^2m)(d^2/dx^2)+(1/2)kx^2]ψ(x)
Kinetic energy Potential energy
Rigid Rotor
This Hamiltonian describes a system where there are two particles whose distance between them are constant and they are rotating about a point. The video above shows a rigid rotor where the flaps are the particles. It is rigid because the distance isn’t changing and it is a rotor because of the rotation. Since the distance between the two particles are always constant, there is no potential energy (V=0). Therefore, the Hamiltonian is solely described by the kinetic energy of the two particles. However, with a rotating rotor, we use angular momentum to determine kinetic energy. As a result, the Hamiltonian becomes:
I = Moment of Inertia
As a result, the Schrödinger equation becomes:
By substituting J for different discrete energy levels, we can we determine the discrete energy levels that excite the rigid rotor. This means that the rotor will spin at only discrete speeds.
Hydrogen Atom Model
In the Hydrogen Atom Model, the electron orbiting the nucleus in the Hydrogen atom is described by a potential equal the Coulomb Potential and a kinetic energy.
Thus, The Hamiltonian of the system is equal to the sum of the Hamiltonian of the kinetic energy and the potential energy of the Coulomb potential (Ĥψ(x)=ψ(x)(ĤRR(θ,φ)+ ĤCP(x))).
By solving this equation we obtain:
Eψ(x)=Ĥψ(x)
Since Hydrogen is a spherical atom, it would make more sense to use spherical coordinates in the Hamiltonian. As a result, the Hamiltonian becomes:
Kinetic Energy Coulomb Potential
The kinetic energy is similar (not exactly the same) to rigid rotor because both operators are given in spherical coordinates.
One possible misconception of the Hydrogen Atom Model is that the orbiting electron could be compared to the Particle in a Box Model and would therefore have a potential energy of 0 Joule. It is true that the electron behaves like a standing wave, resembling the Particle in a Box Model, but it actually has a potential energy that is equal to the Coulomb Potential.
Example Questions:
1. Regarding the particle in a box model, as the energy level approaches infinity, what can you consider the particle? In other words, as n-->infinity, quantum -->??
2. How do the Hamiltonian of the kinetic energy equation in the Hydrogen atom and the Hamiltonian of the kinetic energy of the Rigid-Rotor Oscillator relate?
3. Using the rigid rotor model, calculate the energy level of the rotational energy of a H2 at n =5. Hint: I = µR^2 where µ is the reduced mass.
Answers:
1. As the energy level approaches infinity, the particle can be considered to be free. The quantum solution will approach the classical solution.
2. Both Hamiltonians are written in spherical coordinates, since they consider the angular momentum of the electron.
3. Bond Radius = 74 pm = 7.4 x 10^-11 m
Reduced Mass = (1.66x10^-27)^2/2(1.66x10^-27) = 8.3x10^-28 kg
E=[(5*6)*(6.626x10^-34)^2] / [8*(3.14)^2(8.3x10^-28)(7.4x10^-11)^2] = 2.716 x 10 ^-30 J
References:
Gottlieb, Eric, Harmonic Oscillator, UC Davis Chem Wiki, 5 May 2013, 16 Sept 2013, <http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Trapped_Particles/Harmonic_Oscillator>
Branson, Jim, The 1D Harmonic Oscillator, Quantum Physics 130, Quantum Mechanics, 22 April 2013, 16 Sept 2013, <http://quantummechanics.ucsd.edu/ph130a/130_notes/node153.html>
Delmar, Quantum Mechanical H Atom, UC Davis Chem Wiki, 10 May 2103, 16 Sept 2013, <http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Trapped_Particles/Atoms/Quantum_Mechanical_H_Atom>
Delmar, Rigid Rotor, UC Davis Chem Wiki, 20 May 2013, 16 Sept 2013, <http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Free_Particles/Rigid_Rotor>
Nave, R., Energy Calculation for Rigid Rotor Molecules, HyperPhysics, 16 Sept 2013, <http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotqm.html>
Nave, R., Particle in a Box, HyperPhysics, 16 Sept 2013, <http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html>
A Free Particle, 16 Sept 2013, <http://electron6.phys.utk.edu/qm1/modules/m1/free_particle.htm>
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