The following 2 pictures of the blackboard record our qualitative discussion of quantum mechanics, carried out in the first hour of the second meeting. The first photo, left part shows the representation of classical mechanics. Newton equation, in 1 dimension, with force obtained as a gradient of the potential energy V, in one dimension simply the derivative.

The purpose of this was to show that we must in most cases solve numerically the differential equation of second order in time t. At the bottom we show how this is most easily done, converting it to a system of two first order equations. Further is sketched how this can be solved numerically by the (not too exact) Newton Euler method.  The periodic curve illustrates a solution for a bound system, the mass point is bound in the potential V and moves back and forth between the limits given by the so called turning points, which are obtained as crossing points of the total energy and the potential energy, i.e. the points where the kinetic energy is zero (the mass point stops there and starts moving the other way).

Concepts to remember

• Newton equations, total energy, potential energy, kinetic energy
• The energy does not appear in Newton equation
• The conservation of energy can be derived, but is not seen explicitely
• Turning points

• Drawing diagrams of energy; bound mass point vs. free mass point
• How to show easily that energy is conserved when potential not function of velocity

Quantum mechanics, on the other side starts from the relation for total energy, the sum T + V. It is not really the energy, but the Hamilton function which is the starting point, but for many system it is the same physical, if not mathematical, quantity. The Hamiltonian (and Lagrangian) formulation of classical mechanics are based on variational methods, i.e. the Lagrange function and the Hamilton function are used to obtain back the equation of motion (calculus of variations).

For our quantum physics the sketch in the right part summarizes quite sufficient knowledge of how the Schrödinger equation is obtained, or rather from which postulates. 

One-dimensional time dependent Schrödinger equation is the last formula shown. Above is a sketch of what an operator is. 

The Schrödinger equation for a wavefunction 'psi' is obtained by this prescription

• take the T+V (or similar), express it using momentum p = mv
• replace p by the operator indicated
• replace energy by the operator  indicated
• operators are here 'take a derivative'  multiplied by Planck constant and the imaginary unit
• apply these operat(ors)ions to the unknown 'psi'
• Schrödinger equation for 'psi' is obtained

The rest of the left part shows how the Schrödinger equation becomes the wave equation - but the one for standing waves.

The interpretation of the square of modulus of the wavefunction as probability density is mentioned, and so imposed requirement of finite integral over the whole space (probability that the mass point is somewhere must be ONE. The types of numerical solutions are indicated, the behaviour at the remote values of distance. For most energies ('negative' compared to threshold for free motion set as zero) the solutions of 'psi' blow up in absolute value. Only for very few values the 'tails' go to zero. These energies and their corresponding wave functions describe the possible bound states.

Thus follow energy levels.

To describe transitions, light emission, etc, we must understand energy exchange between subsystems. Light emission is transfer of energy from the atomic motion to the electromagnetic field. No 'jumps' occur. Auger process mentioned, the energy is taken away by another electron, not by electromagnetic pulse, called photon.

This completed the discussion of bound states, transition, light emission, etc.

The last wavy drawing shows the positive energy case, i.e. the quantal description of the scattering events of mass points by potential energies. This indicates how to solve the problem, and how it is related to the wave equations for general wave motion.