(b) = 0.18 cm In this case the wavelength is significant. While the De Broglie equation applies to all systems, the wave properties become observable only for microscopic objects.

An application of De Broglie’s idea

An application of De Broglie’s idea A key idea from optics: You can “see” an object about ½ the size of the wavelength of light used.

An application of De Broglie’s idea A key idea from optics: You can “see” an object about ½ the size of the wavelength of light used. So for visible light at 400 nm, you could see an object with a size of around 200 nm or 2x10-5cm.

An application of De Broglie’s idea A key idea from optics: You can “see” an object about ½ the size of the wavelength of light used. So for visible light at 400 nm, you could see an object with a size of around 200 nm or 2x10-5cm. To “see” something smaller, could use X-rays – but its hard to focus X-rays.

However, could work with electrons in place of light.

However, could work with electrons in place of light. Relatively easy to accelerate electrons to high velocity, so from the formula above, the wavelength would be small.

However, could work with electrons in place of light. Relatively easy to accelerate electrons to high velocity, so from the formula above, the wavelength would be small. Can now see individual atoms.

The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle In 1927 Heisenberg proposed a principle of great importance in the philosophical groundwork of quantum theory.

The Heisenberg Uncertainty Principle In 1927 Heisenberg proposed a principle of great importance in the philosophical groundwork of quantum theory. Heisenberg’s principle is primarily concerned with measurements on the atomic scale.

In classical physics it was generally believed that the accuracy with which any quantity could be determined depended upon the instrument used to do the measurements.

In classical physics it was generally believed that the accuracy with which any quantity could be determined depended upon the instrument used to do the measurements. Thought experiment: Suppose we wish to monitor the position and velocity of a tennis ball in flight. Can do this with different instruments. Important consideration – must be able to “see” the tennis ball – simply shine light on the ball.

Because photons (in the visible part of the electromagnetic spectrum) have small momentum, the impact they have on the tennis ball is negligibly small.

Because photons (in the visible part of the electromagnetic spectrum) have small momentum, the impact they have on the tennis ball is negligibly small. Consider the corresponding experiment on an electron in flight. Must be able to “see” the electron. This requires light with a very small wavelength. Recall the two equations:

Because photons (in the visible part of the electromagnetic spectrum) have small momentum, the impact they have on the tennis ball is negligibly small. Consider the corresponding experiment on an electron in flight. Must be able to “see” the electron. This requires light with a very small wavelength. Recall the two equations: E = h and hence .

Because photons (in the visible part of the electromagnetic spectrum) have small momentum, the impact they have on the tennis ball is negligibly small. Consider the corresponding experiment on an electron in flight. Must be able to “see” the electron. This requires light with a very small wavelength. Recall the two equations: E = h and hence . Therefore the light has high energy if the wavelength is very small.

The impact of high energy photons on an electron can appreciably alter the flight path of the electron. That is, the electron’s velocity and its momentum are altered.

The impact of high energy photons on an electron can appreciably alter the flight path of the electron. That is, the electron’s velocity and its momentum are altered. Thus at the very instant we determine the position of the electron – we lose information on its momentum.

The impact of high energy photons on an electron can appreciably alter the flight path of the electron. That is, the electron’s velocity and its momentum are altered. Thus at the very instant we determine the position of the electron – we lose information on its momentum. To reduce the impact we could choose light of a longer wavelength (i.e. lower energy), but then we would be unable to pinpoint the position of the electron.

This kind of reasoning led Heisenberg to the Heisenberg Uncertainty Principle:

This kind of reasoning led Heisenberg to the Heisenberg Uncertainty Principle: It is impossible to determine to arbitrary accuracy both the position and momentum of a particle at the same time.

This kind of reasoning led Heisenberg to the Heisenberg Uncertainty Principle: It is impossible to determine to arbitrary accuracy both the position and momentum of a particle at the same time. Let x represent the position and p the momentum of a particle. Let and denote the uncertainties in the measurements (i.e. the measurement errors), then

Heisenberg showed that

Heisenberg showed that Of course it always possible to do a sloppy experiment, then the product of the two uncertainties could be a lot larger than .

Heisenberg showed that Of course it always possible to do a sloppy experiment, then the product of the two uncertainties could be a lot larger than . Like the De Broglie relation, the Heisenberg Uncertainty Principle applies to both macroscopic and microscopic objects.

The Heisenberg Uncertainty Principle sets an inherent upper limit for measuring sub-microscopic objects.

The Heisenberg Uncertainty Principle sets an inherent upper limit for measuring sub-microscopic objects. Consequently, it is impossible to pin down the exact nature of the electron.

The Heisenberg Uncertainty Principle sets an inherent upper limit for measuring sub-microscopic objects. Consequently, it is impossible to pin down the exact nature of the electron. Whether an electron should be thought of as a particle, a wave, or a particle-wave – depends on how we take a measurement.

When we speak of size, mass, and charge of an electron – we are thinking about its particle properties.

When we speak of size, mass, and charge of an electron – we are thinking about its particle properties. When we speak of wavelength of an electron – we are talking about the electron as a wave.

When we speak of size, mass, and charge of an electron – we are thinking about its particle properties. When we speak of wavelength of an electron – we are talking about the electron as a wave. There are some additional properties of the electron that we will encounter that will tie in directly with the wave-like nature of the electron.

Problem Example, Heisenberg Uncertainty Principle: The uncertainty in measuring the velocity of an electron is 1.0 x 10-6 cm s-1. Calculate the corresponding uncertainty in determining its position.

Problem Example, Heisenberg Uncertainty Principle: The uncertainty in measuring the velocity of an electron is 1.0 x 10-6 cm s-1. Calculate the corresponding uncertainty in determining its position. Now p = m v

Problem Example, Heisenberg Uncertainty Principle: The uncertainty in measuring the velocity of an electron is 1.0 x 10-6 cm s-1. Calculate the corresponding uncertainty in determining its position. Now p = m v so = m

Problem Example, Heisenberg Uncertainty Principle: The uncertainty in measuring the velocity of an electron is 1.0 x 10-6 cm s-1. Calculate the corresponding uncertainty in determining its position. Now p = m v so = m = (9. 11 x 10-28 g)(1.0 x 10-6 cm s-1) = 9.1 x 10-34 g cm s-1

From , for the purpose of calculation, we can treat the as an = sign From , for the purpose of calculation, we can treat the as an = sign. Hence,

From , for the purpose of calculation, we can treat the as an = sign From , for the purpose of calculation, we can treat the as an = sign. Hence,

From , for the purpose of calculation, we can treat the as an = sign From , for the purpose of calculation, we can treat the as an = sign. Hence, = 5.8 x 103 m (or 3.6 miles!)

Exercise: Try a similar calculation on your instructor (or yourself) Exercise: Try a similar calculation on your instructor (or yourself)! Assume a mass of around 80 kg (or your mass) and an uncertainty in velocity of around 0.5 cm s-1. Is the uncertainty in the instructor’s position (your position) so uncertain that he (you) is (are) not in the room?

Quantum Mechanics The spectacular initial success of Bohr’s theory was followed by a series of disappointments.

Quantum Mechanics The spectacular initial success of Bohr’s theory was followed by a series of disappointments. Bohr (nor anyone else) could account for the emission spectra of atoms more complex than the H atom.

It was realized eventually that a new theory was required It was realized eventually that a new theory was required. A new equation beyond Newton’s laws of motion was needed to describe the microscopic world.

It was realized eventually that a new theory was required It was realized eventually that a new theory was required. A new equation beyond Newton’s laws of motion was needed to describe the microscopic world. The key figure to make the great advance was Schrödinger (1926).

It was realized eventually that a new theory was required It was realized eventually that a new theory was required. A new equation beyond Newton’s laws of motion was needed to describe the microscopic world. The key figure to make the great advance was Schrödinger (1926). Schrödinger formulated a new equation to describe the motion of microscopic bodies.

Schrödinger formulated an equation that incorporated the particle properties in terms of the mass (m) and charge (e) and the wave properties in terms of a function – which is called the wave function. The symbol employed for the wave function is (psi).

Schrödinger formulated an equation that incorporated the particle properties in terms of the mass (m) and charge (e) and the wave properties in terms of a function – which is called the wave function. The symbol employed for the wave function is (psi). There is no simple physical meaning for the function . However the function has the meaning that it gives the probability of finding the particle at a particular point in space.

This probability thinking is a radical break with classical thinking.

The Hydrogen atom and the Schrödinger equation

The Hydrogen atom and the Schrödinger equation One of the first problems investigated by Schrödinger was the energy levels for the H atom.

The Hydrogen atom and the Schrödinger equation One of the first problems investigated by Schrödinger was the energy levels for the H atom. Schrödinger obtained the result:

where: m = the mass of the electron

where: m = the mass of the electron e = charge on the electron

where: m = the mass of the electron e = charge on the electron h = Planck’s constant

where: m = the mass of the electron e = charge on the electron h = Planck’s constant n = a positive integer value ( n = 1, 2, 3, …) called the principal quantum number

where: m = the mass of the electron e = charge on the electron h = Planck’s constant n = a positive integer value ( n = 1, 2, 3, …) called the principal quantum number = vacuum permittivity

where: m = the mass of the electron e = charge on the electron h = Planck’s constant n = a positive integer value ( n = 1, 2, 3, …) called the principal quantum number = vacuum permittivity En = the energy of the different states

This formula shows that the energies of an electron in the hydrogen atom are quantized.

The above formula will be abbreviated as k En = – n2 This formula shows that the energies of an electron in the hydrogen atom are quantized. The above formula will be abbreviated as k En = – n2