From the previous discussion on the double slit experiment on electron we found that unlike a particle in classical mechanics we cannot describe the trajectory of an electron. We can however associate a wavefunction with each electron which will tell us the probability amplitude of finding out an electron at a given point in space at a given point in time.This wave function or probability amplitude has to be complex to account for the double slit interference pattern created on the screen since the double slit pattern is not a simple scalar some of the pattern coming out of the individual slit.We shall denote this probability amplitude or wave function as Ψ(x,t) The exact functional form of this wave function will depend on the potential under which the electron is moving.Also it is now clear that since we cannot determine the trajectory of a single electron in Quantum mechanics we shall not be interested in finding out the time derivative of position or time derivative of velocity. But we shall be interested in finding out the time derivative of the wavefunction it self namely dΨ/dt. Thus we shall try to figure out which law calculate this quantity.

Position and Momentum From Classical Mechanics we find that to know all physical quantities we need to find out the position and momentum. In Quantum mechanics correspondingly we need to find out the probability distribution of position and momentum. In the previous section we found out that the wavefunction gives us the probability of finding a quantum mechanical particle at a given point of space at a given point time. What about the momentum? Let us take a special candidate for such wave function namely plane wave something you have already studied in E.M. theory. In one spatial dimension such a wave function is written as Ψ(x,t)=e ikx where k is the wavenumber. The probability density gives the intensity of the electron wave on the screen is |Ψ(x,t)| 2 =1.This means that the probability is uniform and the position is completely uncertain. Remember we are talking of the same electron and after coming through a given slit under any condition they can end up anywhere on the screen.

Momentum of a plane wave The wave number associated with such wave function is k. According to the de Broglie hypothesis then the momentum associated with electron is simply given by p= ħk. This is a remarkable result. It just shows that while the postion of electron given by such wave function is completely uncertain it has a definite momentum. Moreover a simple mathematical trick tells us that that above result can be written as in the following way -iħ∂Ψ(x,t)/∂x = ħkΨ(x,t)= pΨ(x,t) The above result tells us something very important. At least for plane wave like wave function if we operate it by a differential operation which is in this case the first spatial derivative we get the same wave function multiplied by the value of its momentum. Thus there exist some connection between the x-component of the momentum p and the differential operator -iħ∂/∂x

Momentum (Contd.) Using a more complete and rigorous mathematical theory which we cannot unfortunately describe in this course ( but generally taught in a full course on Quantum Mechanics) actually it can be shown that indeed -iħ∂/∂x represents the momentum operator along x-direction. The above statements means, for example if we want to know the momentum associated with the wavefunction ( probability amplitude) of an electron we need to operate the function with the above operator. This brings us to a strange fact again which we need to interpret. Suppose the wavefunction of the electron is not given by a single plane wave but a combination of many such plane wave. There is theorem in mathematics which you may know. This is called Fourier theorem. It actually tells that any well behaved function can be created by adding up a large no. of plane waves with different values of wave number ( or wave vector) k. We shall particularly consider two such cases Ψ(x,t)=e ik 1 x+ik 2 x and Ψ(x,t)=Aexp(-x 2 /σ²).

Momentum eigenfunction In both cases if we operate the wave functions with the operator -iħ∂/∂x we do not get the same wave function multiplied by a definite value of the momentum back, namely -iħ∂Ψ(x,t)/∂x ≠ pΨ(x,t). What will be the momentum of the electrons with which such wavefunctions can be associated? The answer to this question is unlike the electron with which we associate a single plane wave like wavefunction, these electrons do not have a defnite wave number k or definite momentum p. They exist actually in a mixture of electronic wavefunctions each of which has a specific momentum (plane wave). In the first example the wave function composes of two such momentum values since it is a combination of two plane waves. In the second case using the mathematical theorem mentioned earlier it can be shown that the wave function is a combination of infinite number of plane waves. When we try to measure the momentum of such electrons we ended up getting these different values of composing momentums at different time and we call their momentum uncertain.

The case of the Gaussian wave function is even more interesting. Here the function is localized within the length . therefore the uncertainty in finding the electron position gets reduced as compared to plane wave. However this state as we have mentioned can be mathematically written summing up a large number of plane waves. Thus its momentum gets very uncertain. A comparison of this wavefunction with the plane wave like wavefunction indicates that if we are able to increase the uncertainty in position by choosing a new wavefunction this results in an increase of uncertainty in momentum We can now summarize the preceding discussion to get some important conclusions. We can associate the following operators with the three component of momentum namely p x  -iħ∂/∂x, p y  -iħ∂/∂y, p z  -iħ∂/∂z.. The wavefunction associated with a quantum mechanical state can be either in a state of definite momentum ( plane wave) or in a state with mixture of various values of momentum giving uncertainty in momentum.

If the wavefunction has a definite momentum the action of the momentum operator will retain the same wavefunction multiplied by the value of that definite momentum. Such a wave function is called the eigenfunction of the corresponding operator ( in this case the momentum operator) and the value of the momentum that multiplies the wavefunction is the momentum eigenvalue. If the wavefunction is not in a definite momentum state, but in a superposition of large number of momentum states, then the action of the momentum operator on such state will yield a different function altogether. The measurement of momentum of this wavefunction will yield different values at different time according to the composition of wavefunction, but each time one will get a specific value of the momentum. The analysis of the wavefunction also indicates that if for a given wavefunction the uncertainty in the position becomes less, the uncertainty in momentum correspondingly increases.

The positional operators are just given by the co-ordinate variables themselves, namely x,y,z In classical mechanics most of the physical quantities ( better known as dynamical variables) such as angular momentum, kinetic energy, potential energy can be written as function of position and momentum variables. In quantum mechanics we can associate an operator with each such dynamical variable by replacing the position and momentum variables with their respective operator. This means The significance of replacing a dynamical variable by a operator is the following. Now if a wavefunction is given operating the wavefunction by the operator corresponding to a given dynamical variable ( such as energy momentum) we can immediately find out if the wavefunction is an eigenfunction of that operator or not.

If the wavefunction is an eigenfunction of that operator then the corresponding particle is going to have a definite value of the related physical quantity. For example, if the wave function of an electron is an eigenfunction of the z- component of the angular momentum operator then it has a definite L z. Same is true for energy, momentum etc. On the otherhand if the corresponding wavefunction is not an eigenfunction then everytime one measures the corresponding physical quantities on the same quantum mechanical particle under identical condition one will end up with different numbers. The corresponding physical quantity becomes uncertain. However we can learn the following things about such uncertain quantities. The wavefunction gives the probability amplitude at a given space and time.The corresponding probability density is given by the modulus square of the wavefunction. Using this probability density we can calculate the mean value and higher moments of any physical quantity associated with the particle in the following way.

The probability density of finding an electron in a small volume element d  around the point r is given by Since the electron or any other quantum mechanical particle must be found somewhere in the space the total probability can be normalized to 1. The expectation value of any dynamical variable is therefore given by Such an expectation value can be obtained by repeating the same experiment for a large no. of times and taking the average. The dynamical variable thus behaves like a random variable.

Variance We know that for any such distribution, we can define other moments, say We define the variance of O as And the standard deviation as This is the spread in the results observed when we make a large number of independent measurements and can be used to quantify the uncertainty in that particular physical quantity. For example if we use the plane wave state as the wave function and calculate the standard deviation in the momentum, we shall find it is 0. Thus the momentum of that state is well defined

Uncertainty Principle We have already pointed out by taking a plane wave like wavefunction and a localized Gaussian type of wavefunction that in the case of the former the position is completely uncertain whereas the momentum is definite and in the later case position has much more uncertainty, but the momentum is no more definite. Explicitly calculating the standard deviation using the preceding formulas the above statement can be easily verified. The message is that uncertainty in momentum and position along a given direction are somewhat inversely related. Around 1925 Heisenberg expresses this fact mathematically by stating we cannot measure a pair of variables like position and the associated momentum with arbitrary accuracy in the same experiment

Heisenberg Microscope

Complementarity Thus Heisenberg’s principle says that quantum mechanics imposes certain limits on the accuracy with which we can observe the world. A pair of variables like position and its associated momentum which we cannot observe accurately together are said to be complementary variables. Other such complementary pairs are rotational angle and the associated angular momentum. The product of each pair has dimensions =[h]!

Commuting Operators If two quantities can be measured simultaneously without disturbing each other, they are said to commute. For instance we can measure the x component of position of an object without disturbing its y component and then measure the y component or vice versa. So we expect Here I have used a crescent symbol to indicate the fact that the x is a measurement of the x coordinate. So if the operators commute we can measure them in any order.

Non-Commuting Operators If two quantities cannot be measured simultaneously without disturbing each other, they do not commute. For instance we can measure the x component of position of an object but that disturbs the corresponding momentum and vice versa. So we expect To achieve this in a way compatible with Heisenberg’s uncertainty principle, we set Thus So we can write The above quantity is called the commutator of x and p x