sign in sign up
<<
Home , Physicist

Lecture XI:I March 3 , 2021

I. : HEISENBERG & SCHRODINGER¨

In (1995), German physicist and Austrian physicist Schrodinger¨ independently developed two distinct but equivalent frameworks to describe microscopic world that replaced Newton’s classical equations with new equations of quantum theory. The new theory is called Quantum Mechanics. As we will see, the equation of quantum mechanics is completely different from described by Newton’s equation. Schodinger¨ theory is also known as “ Wave Mechanics”. Both theories produced the results of the Bohr model, but also explained the intensities of and provided insight into what is meant by de Broglie wave. Heisenberg won Nobel prize in 1932 and Schodinger¨ won the Nobel prize in 1933.

II. HEISENBERG’S QUANTUM THEORY

• We cannot see Bohr orbits and only thing we can measure is the radiations coming from ... Therefore, light emitted from the encodes all the secrets of Bohr theory...

Heisenberg tried to find a sort of bookkeeping method for all possible atomic transitions and this lead him to matrices – square arrays of numbers. In quantum , each number represents a possible atomic transition.

He formulated quantum theory– equations of quantum physics in terms of matrices..

Heisenberg’s strange : a × b 6= b × a .

  T11 T12 T13 T14....    T T T T ...   21 22 23 24  M =   . (1)  T T T T ...   31 32 33 34  T41 T42 T43 T44...

Here T11 = T22 = T33 = T44 = 0 and T12 represents the transition from n = 1 and

1 n = 2 and so on... Heisenberg calculated energies of an atom and the atomic transitions for emission or absorption of light when atom jumps from one orbit to another.

• Even the simple classical things like position of a particle, its velocity were represented as matrices in Heisenberg’s theory.

• Given two matrices A and B, A × B 6= B × A. Therefore, position x and momentum p are such that x × p 6= p × x.

• The matrix quantum theory led to Heisenberg uncertainty principle – we cannot measure the position and momentum of the particle simultaneously... ∆x∆p ≥ ~/2. This is known as Heisenberg Uncertainty Principle. This explains why electron cannot spiral down to the nucleus as electron at rest will have infinite energy.

• used Heisenberg’s matrix equations to derive the Bohr model... We will hear about Pauli later...

Heisenberg theory provided clear evidence that quantum world is very different from the classical world. Schrodinger¨ Quantum Mechanics: Wave Mechanics

In December 1925 while on vacation, Schrodinger¨ ( Univ of Zurich physics professor ), looked at de Broglie’s thesis. He worked out a single equation, explaining the behavior of particles in terms of de Broglie waves. The lead player in the equation is a quantity called Ψ ( pronounced ”sigh” ) which is called the wave function.

• Instead of describing particle by its position and velocity, in Schrodinger’s¨ equation, the particle is described by wave function Ψ.

• Even in , waves are described by wave functions, which gives the amplitude, wavelength and shape of the wave. However, Schrodinger¨ wave function is not a “real” quantity. In other words, unlike water or sound or even electro-magnetic waves, matter waves are not described by ordinary real numbers. Since it is not real , called complex number, we cannot determine the shape of the wave from its wave function.

2 • Known as the Schrodinger¨ equation (SE), this partial for a non-relativistic particle of mass m in a potential V is given by

∂ − 2  i Ψ = ~ ∇2 + V (r, t) Ψ (2) ~∂t 2m √ Here i is the complex number equal to −1. The function Ψ, known as the wave function of the particle, encodes complete information about the state of the particle .

• Schrodinger¨ equation determines the energy and predicts the atomic orbits identical to that of Bohr model...if we interpret the |Ψ|2 as the probability. It shows only certain values of 2 energy are permissible and have the form En = 13.6/n .

BIG QUESTION: What is Ψ ?

Schodinger¨ himself did not know what Ψ was ? While Ψ represents some kind of wave,

what exactly was this wave ?? What is waving ??

Max Born: Nobel prize 1954

It was , who successfully interpreted the wave function Ψ as the probability amplitude of the wave associated with the particle and was awarded the Nobel prize in 1954.

Unlike classical wave functions, Ψ is a complex number and is not altogether a measurable quantity. Therefore, unlike water waves, or waves in a string, or electromagnetic waves, where the wave function is an observable entity describing oscillations of the medium or electromagnetic fields, the wave function for a is an abstract quantity .

Max Bohr provided physical interpretation of the Schrodiner¨ equation function – its square gives the probability of finding electrons. Therefore, matter waves are probability waves. That is, when we draw a wave describing a quantum particle, what we are showing is the square of the

3 wave function that comes from the Schrodinger¨ theory.

It is the absolute square , namely |Ψ(r, t)|2, that is a physical entity describing the probability of finding the particle at location r at time t.

While the probability amplitude encodes all the information about the state of the particle, taking the absolute value (the modulus) destroys some information (called the phase). This subtle distinction is the ultimate source of all quantum mechanical “weirdness”.

HOW TO UNDERSTAND BOHR MODEL in terms of Sch”odinger Equation ???

Correct picture is: There are NO circular orbits. Electron can be any where. However, there are some points where the probability of finding electrons is highest. Magically, these points coincide with the circular orbits.

Reconciling de Broglie, Bohr, Heisenberg and Schrodinger¨

Before 1925, there was Bohr model and the de Broglie wave theories that describe some aspects of quantum particles. However, there was no general theory, something that will be the h analog of Newton’s equations of motion. Although de Broglie gave the equations λ = p and E f = h , it did not say what kind of wave is associated with the quantum particles and what is the equation that describes propagation of these waves. The key unknown was : if electrons are wave, what is the wave equation, just like there is a wave equation that describes sound waves and Maxwell’s equation that describes electromagnetic waves. Bohr model focussed on stability of atom and explained how radiations are emitted from atomic gas. But, did not tell us, why some colors ( wave lengths ) are stronger than others...

In 1926, there were two theories – Heisenberg and Schrodinger¨ theory that explained the behavior of quantum particle completely, just the way the Newton’s equation F = ma explains the behavior of classical particles. Heisenberg model focussed on the transition probabilities that give rise to various colors from the atoms. The results agree completely with the Bohr model. It also gave additional information

4 about the intensities of various colors and thus explained why some lines in the spectrum are more intense than the others. Scrodinger¨ gave us a wave-like equation that describes the motion of quantum particles. Solutions of the equation are complex numbers, but it calculated the energies of the electron in H-atom and found it to be in complete agreement with Bohr model. It also explained the intensities of the spectrum lines. The revolutionary idea emerged is that we CANNOT determine where the particle is. All we can determine is the probability for a particle to be at any location. Thus unlike the classical equations (Newton’s laws) that predict where the particle is, the quantum equation predicts only the probability of where the particle is. In other words, rules of nature are unpredictable, just like the rule for determine head or tail in the tossing of the coin

5