Lecture 25: Wave mechanics

Review

Notions of quantum mechanics

  • Energy is quantized
  • Light can be thought of as a particle
  • Electron can be thought of as a Wave
  • Wave-particle duality
  • Location of a particle, how certain can we be?

Today

  • Heisenberg’s Uncertainty principle
  • Waves and their mathematical description
  • Wave equation: Harmonic oscillator
  • Application to quantum mechanics
  • Concept of operators
  • Schrodinger’s equation
  • Particle in the Box problem

Heisenberg’s Uncertainty principle

Heseinberg, a student of Niels Bohr, recognized the true consequence of wave-particle duality. He reasoned that if we were to describe position and momentum of “quantum size” particle it will be difficult to measure both the quantities simultaneously. That is, the act of measuring position of a particle, say by shining a light on it, will influence the momentum of the particle creating uncertainty in both the quantities. He showed that:

The implications of the simple equation are profound. It took away the infinite certainty implicit in the Newton’s description of motion and position of a particle. That is, in classical mechanics, we can “measure” the position and location of the particle with any degree of certainty using ever increasing sophisticated techniques.

When Einstein learned about the Heisenberg’s theory, he was deeply saddened. He is claimed to have said “ God does not play dice with Men.” However, the profound truth of the above equation endured despite Einstein and other physicists attempts to find some flaw in the reasoning that lead to the above equation.

Since the quantum particle cannot located with arbitrary precision, we must describe it in terms of it’s probability of finding at a given position.

Wave mechanics

Thanks to De Broglie, we know that particles can act like waves. So it is interesting review some basic concepts about waves. Consider an example of rolling hills; we may sketch them as follows.

Note how the structure is periodic. If we want to describe topological height as a function of distance, we can employ a simple trigonometric sine function:

If we want to change the periodicity we can simply substitute for variable Distance = k.Distance. As we increase the magnitude of k, the period between the oscillation decreases and vice versa. The periodicity is best described by the wavelength, =2/k. So higher the k lower is the periodicity. So for a space dependent wave we may write:

A is the amplitude, and  is the period of the wave. Just as wave can occur in space, there can be many properties that can have a wave-like behavior in time, for example sound waves.

Harmonic Oscillator

Consider a small weight attached to a mass-less spring. If we pull the weight, stretching the spring, and release it, the attached mass will compress and expand the spring following a periodic motion. How do we describe the motion of the mass in this case? We use the Newton’s law:

Thus motion of a harmonic oscillator is sinusoidal in time. Comparing this equation with the space dependent variation equation:

where a=2/=k/m=oscillation frequency(rad/sec) and 1/=(frequency,Hz)

Notion of standing waves

Consider a string whose both ends are attached to solid support, like a guitar. If we stretch it and release it, then only certain wavelengths are allowed. This is because at the tied ends the amplitude waves must be zero.

Thus notion of Quantization of wavelength arises, even naturally in classical problems. This was exploited by Schrodinger develop a famous equation that bears his name.

Wave mechanics

As we have seen, de Broglie suggested that any particle has wavelength given by =h/mv. Thus, electron could also be treated as wave. For example consider following problem.

Thus the electron wavelength is comparable to typical molecular bond length! To describe the motion as well as location of this electron wave we must develop an equation like Newton’s force =ma equation. How can we use Bohr’s picture to define the electron as a wave in it’s own orbit?

Electron as a standing wave in Orbit

Schrodinger equation

Schrodinger began developing the theory of electron waves with a classical general time and space dependent equation of a wave.

where c is the velocity of the wave.  is a function that depends on both time and distance. The solution of this equation can be written as:

Note here we have 2v=a, and De Moivre’s theorem identity gives us:

Thus the solutions (wavefunctions) are sums of sine and cosine functions but for algebric reason it is useful to maintain complex exponential notation. The key insight is to recognize we can substitute for electron wavelength,=h/p, and for frequency =E/h


Schrodinger equation (continued)

Similarly we can show by differentiating with respect to position.

Note the last equation defines entirely new concept; the momentum is an operator. He made use of this connection to describe the total energy of electron system, which is simply sum of potential and kinetic energy.

Whence, we can write