Wave Mechanics

Wavelike Properties:  Traveling Waves vs. Standing Waves

In the most general sense, waves are particles or other media with wavelike properties and structure (presence of crests and troughs) which can be mathematically described by a wavefunction or amplitude function. From this broad definition, one can categorize “waves” into two different groups: traveling waves and stationary waves. Traveling waves, such as ocean waves or electromagnetic radiation, are waves which “move,” meaning that they have a frequency and are propagated through time and space. Another way of describing this property of “wave movement” is in terms of energy transmission – a wave travels, or transmits energy, over a set distance.

However, stationary waves, or standing waves, remain in a constant position. Unlike a traveling wave, the crests and troughs of standing waves are in fixed intervals separated by nodes (regions of the wave where there is no movement). One way of producing a variety of standing waves is by plucking a melody on a set of guitar or violin strings. When placing one’s finger on a part of the string and then plucking it with another, one has created a standing wave. The resulting vibration, the up and down displacement of the string, creates the crests and troughs of the standing wave. And, the point at which one’s finger is holding the string in place is the node (n) of the standing wave. 

Standing Waves as Wave Functions

As was briefly mentioned earlier, the concept of a wave function, denoted as psi (or Ψ), was developed by Erwin Schrödinger as a mathematical description (or equation) of the wavelike properties of particles and many other types of media, both micro- and macroscopic. The actual equation of a wave function is called the Schrödinger Equation. However, for the scope of this exercise, we will only be discussing the main concepts of the wave function in order to describe the behavior and properties of a standing wave.  

Relating Quantum Number (n) of a Wave Function (Ψn) to Kinetic Energy:

Just as gas molecules, photons, and electrons have kinetic energies, a free moving particle also has a kinetic energy, E_k, which can be expressed in wavelengths λ (using de Broglie's wavelength-momentum equation) of matter waves (Ψn). The resulting simplified equation, E_k =n²h²/8mL²  reveals several key properties of standing waves: quantized energy, the zero-point energy, and the Heisenberg Uncertainty Principle. First, a standing wave’s energy E_k is quantized, meaning that the wave’s energy exists as only in discrete values determined by the number of nodes or quantum number n = 1,2,3 … etc. Second, a standing wave particle cannot have a zero-point energy, n = 0. The particle must always have some form of energy and cannot exist at rest, therefore, neither the particle’s energy values (E_k) nor can the quantum number (n) equal zero. Third, the relationship between the particle’s kinetic energy and the length of the one dimensional box in equation E_k =n²h²/8mL²=p²/2m also confirms the Heisenberg Uncertainty principle: as one decreases the length of the box, L, one increases the particle’s kinetic energy; and, thus, according to Heisenberg’s Uncertainty principle, the increase in a particle’s kinetic energy decreases our ability to determine the particle’s momentum.   

1. The kinetic energy of a free moving particle has the equation: E_k = mu²/2 = m²u²/2m= p²/2m, where m is the mass of the particle, u is the speed of the particle, and p is themomentum of the particle. 

Kinetic Energy of a particle:   

E_k = mu²/2 = m²u²/2m = p²/2m

1. One can then translate the particle’s kinetic energy in terms of wavelength by substituting the particle’s momentum, p, with de Broglie’s wavelength-momentum relationship, λ = h/p, where λ is the particle’s wavelength, p is its momentum, and h is Plank’s constant 6.62607 x 10^-34 J s.

De Broglie’s wavelength-momentum relationship:    

λ = h/p  -->  p = h/λ

Substituting the wavelength equation into the kinetic energy equation:  

E_k = p²/2m = h²/2mλ²

1. Lastly, one can further express this kinetic energy-wavelength equation as a “matter wave” or standing wave by substituting λ with the standing wave definition of a wavelength, λ = 2L/n, where L is the length of the one dimensional box and n is the number of nodes, or principle quantum number. It is important to note that n is equivalent to sum of all crests and troughs located within the length, L, not the total number of nodes.  

Standing wave definition of wavelength:    

λ = 2L/n

Substituting the standing wave’s wavelength equation into the modified kinetic energy equation:

E_k = p²/2m = h²/2mλ² =h²/2m(2L/n)² = n²h²/8mL²

Quantum Numbers and Electron Charge Density

Now that we have used the wave function to describe the general relationships between a given particle’s quantum number (n), kinetic energy (E_k), and wavelength (λ), what do the different values of the wave function represent? And, how is the behavior of the wave function graph related to other aspects of quantum mechanics?

Interpreting a wave function (Ψ): positive, negative, and zero values

Whether confined to the volume of an atom or the dimensions of a “particle in a box,” an electron is a minute particle which displays wavelike properties and, therefore, has a corresponding wave function (Ψ). The electron’s wave function values represent energy quantities which can be positive, negative, or zero (no energy or movement) depending on its position along the wave function.

One might then ask, what is the physical significance of these positive and negative energy values? None – the wave function of an electron has no a physical significance because an electron behaves as a wavelike particle, a property of quantum physics, not classical physics. Thus, in order to understand the spatial behavior of wavelike particles, such as electrons and photons, we must manipulate the wave function (Ψ) in terms of electron position probability instead of electron energy. This modified wave function is the square of the wave function (Ψ) and is called the electron charge density (Ψ_n²). It is used to calculate the probability of finding an electron or particle at a particular position on wave function.  

Determining the position or location of an electron

As was stated in the previous section, the electron charge density (Ψ_n²) as well as photon density is a function describing the probability of an electron’s or photon’s position within a given set of dimensions. And, by mathematical definition, the electron charge density(Ψ_n²) is an intensity function, meaning it is an equation which measures the density and quantity of electrons. Thus, the total probability of finding an electron at a particular position within a small volume of space, such as an atom, is the product of the electron charge density and the volume of space.

An electron as a particle in a one dimensional box: particle in a box

The probability of finding an electron in a one dimensional box depends on the number of nodes (n values). At n=1, electron charge density(Ψ₁²), one is most likely to find the electron at the graph’s maximum, the center of the box. At n=2 and n=3, electron charge density (Ψ₂²) and (Ψ₃²) respectively, one again most likely to find the electron at the graph’s maxima; however, there is no probability of finding the electron at the nodes, or zero values, on the graph.

An electron as a particle in a volume (three dimensional box): electron in an atom

However, if an electron is placed in a three dimensional box it is able to spatially move in all three directions within a given volume. To then determine the probability of finding the electron at a particular position in this complex system, one must integrate two more quantum numbers (ℓ and m_ℓ) for each addition dimension into the electron charge density equation (Ψ_n²).  

References

1. 1. Petrucci, RH., Hardwood, WS., Herring, FG., Madura, JD. (2007). General chemistry principles and modern applications. US: Pearson Education Inc.
2. 2. Anslyn, EV., Dougherty, DA. (2006). Modern Physical Organic Chemistry. US: University Science Books.

Outside Links

• http://en.wikipedia.org/wiki/Standing_wave
• http://www.spaceandmotion.com/Physic...wave.equations
• http://physics.info/waves-standing/

Practice Problems

1. What is the length of a string that has a standing wave with 3 nodes (including nodes) and λ=26 cm?

2. What is the wavelength (λ) of a standing wave with n=3?
3. How many nodes (n) can a standing wave with a wavelength of λ=2L have?
4. What part of this trigonometric equation Ψ_n (x) = (L)^(1/2) cos[(2πn)/L]determines the (a) amplitude, (b) period/ spacing of nodes of a wave function? (c) What type of wave function does this equation represent?
5. Is the wave function Ψ_n(x) = (5/L)^(1/2) sin[(xπn)/L]positive or negative at n=3 and x=L/2 ? 

Contributors

• Saarah Kuzay
• Samantha Ma

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