Collected from: Hyperphysics.phy-astr.gsu.edu
http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.html#top
http://outreach.atnf.csiro.au/education/senior/astrophysics/spectroscopyhow.html
http://www.physics.uiowa.edu/~umallik/adventure/quantumwave.html
http://norbert.mit.edu/reactor/bragg_theory.html
http://www.phys.unsw.edu.au/~sjc/physics1/PHYS1221/2007.html
Light EM Wave
At the beginning of the 20th century everyone thought light was a wave. For most purposes physicists still think of light as an electromagnetic wave traveling at 186,282 miles per second!
The different colors are just waves of different frequencies (most frequencies are invisible to our eyes).
Q: Which experiment could not be explained by wave model?
Bohr model explain Hydrogen atomic spectrum. Pretty impressive, but
Doesn’t work for anything but hydrogen atom (hydrogen like atom).
Doesn’t even work when things get more subtle in hydrogen, e.g. Zeeman splitting due to applied magnetic field.
Semi-classical, not a true Quantum theory (truly speaking electrons do not have well defined orbits)
the
Photo electric effect:
The details of the photoelectric effect were in direct contradiction to the expectations of very well developed classical physics.
The explanation marked one of the major steps toward quantum theory.
The second important property of light is interference. If two identical waves go through each other, then their intersection will look like the sum of the two waves. Recall that all waves are fluctuations in something. A typical wave quickly alternates between a fluctuation up and a fluctuation down. That's why, in the above drawing, we represent the wave with alternating black and white lines. The black lines represent upward fluctuations and the white lines represent downward fluctuations. If two intersecting waves both happen to be fluctuating up, then the sum will be a fluctuation up with twice the amplitude. This is called constructive interference. If one is fluctuating up while the other is fluctuating downwards, they will cancel each other out. This is called destructive interference. Return to the Double Slit Experiment Now that we have an idea of how waves behave, we can now predict the results of the double slit experiment. Some of the light will go through slit 1, and some through slit 2. After going through the slit the light will spread out in all directions. The light that went through slit 1 will interfere with the light that went through slit 2. How can we tell from the diagram where the light will interfere constructively and destructively? Well, the light interferes constructively whenever both waves fluctuate up in the same place and time. The light interferes destructively when the waves are fluctuating in opposite directions at the same time. To make this clearer, I've shown the locations of constructive (red lines) and destructive interference (blue lines) in the picture below. W
WWW
We
The result? We
We can see the things, because of light?
Planck curve for a black body at 6,000 K
If you look closely at the curve you will notice that the object emits some radiation at every wavelength including in the ultraviolet and infrared wavebands. You should also notice that the amount of energy emitted is not the same for all wavelengths and that in this case, the peak wavelength falls within the region of visible light. Now what happens if the temperature of the black body source is different? The plot below shows Planck curves for an object at four different temperatures from 6,000 K to 4,000 K. Note the wavelength here is expressed in units of Ångstroms. 1 Ångstrom = 0.1 nanometers.
Credit: Plot generated from an applet courtesy of Mike Guidry
Planck curves for black bodies of differing temperatures.
Albert Einstein extended Planck's work to the light that had been emitted. At a time when everyone agreed that light was a wave (and therefore continuous), Einstein suggested that it behaved as if it was a stream of small bundles, or packets, of energy. In other words, light was also quantized. Einstein's model was based on two assumptions. First, he assumed that light was composed of photons, which are small, discrete bundles of energy. Second, he assumed that the energy of a photon is proportional to its frequency.
E = hv
In this equation, h is a constant known as Planck's constant, which is equal to 6.626 x 10-34 J-s.
The Bohr Model of the Atom
Just, look at a pencil sketch drawn by Humna !!!!!!!!!!!!!
Can you see it?
This can be due to
A) Wave nature only B) particle nature only
C) Both natures simultaneously D) wave or particle nature
y see the spots where the light interferes constructively, and not destructively. Therefore, we willd Wave-Particle Duality: Light
Does light consist of particles or waves? When one focuses upon the different types of phenomena observed with light, a strong case can be built for a wave picture:
Interference Diffraction Polarization
Phenomenon
Can be explained in terms of waves.
Can be explained in terms of particles. Reflection Refraction Interference Diffraction Polarization Photoelectric effect
The particle properties of electrons were well documented when the DeBroglie hypothesis and the subsequent experiments by Davisson and Germer established the wave nature of the electron.
Q: If a wave (light) can behave like a particle, why not vise versa?
Ans: De Broglie Atom: electrons behave as wave. Orbits only exist where standing wave closes. i.e. where is wave length of electron-wave.
Q: Explain the Wave nature of matter?
Ans: According to Louis de Broglie, If PHOTONS can have both wave and particle characteristics then perhaps all other forms of matter do too.
DeBroglie Hypothesis
Suggested by De Broglie in about 1923, the path to the wavelength expression for a particle is by analogy to the momentum of a photon. Starting with the Einstein formula:
Another way of expressing this is
Therefore, for a particle of zero rest mass
For a photon:
The momentum-wavelength relationship for a photon can then be derived and this DeBroglie wavelength relationship applies to other particles as well.
Confirmation of the DeBroglie hypothesis came in the
Grate experiments of PHYSICS
Davisson-Germer Experiment
We shall first need to know some things about crystals and the wave interference effects that may be observed with them. As mentioned above, a crystal represents a collection of many atoms bound together by interatomic forces to form a threedimensional solid (however, twodimensional cases and liquid crystals are known to exist). In a perfect crystal, the atoms are positioned in a spatial array (or lattice) with precision, this being determined by the symmetry and balancing of interatomic forces on each atom. With this regularity of position, illustrated schematically in Figure 4, it is easy to envision the overall crystal as being made up of parallel sheets (or planes) of atoms which can serve to provide wave interference between the components of radiation scattered by individual atoms. Almost any textbook on general or modern physics (e.g., reference [8, 9])will have an elementary derivation of the conditions necessary for constructive interference of radiation scattered from atomsin a crystal plane.These conditions are expressed by Bragg’s Law,
nλ = 2d sin θB,
where n = order of diffraction (1, 2, 3,. . .), λ is the wavelength, d is the interplanar spacing, and θB, called the Bragg angle, is the grazing angle of incidence and reflection. It is deceptively similar in appearance to the law describing constructive interference from a onedimensional set of scatterng centers (e.g. a grating), and it is worth pointing out the difference.
In the one-dimensional case, illustrated in Figure 5, the approach angle θ1 may have any value, and the exiting angle θ2 for constructive interference is then defined by the interference equation with θ2 not necessarily equal to θ1. For reasons not so obvious, this generality is not present in three dimensional diffraction where θ1 = θ2 = θB. In fact, diffraction from a crystal always occurs in symmetrical fashion from atom planes with both incident and emergent angles beingequal to θB. (This subtle distinction is discussed in[10] and in various texts on crystallography, e.g., XRay Crystallography, M. Wolfson, Cambridge 1970.) A given set of atom planes of spacing d will reflect radiation of wavelength λ with intensity concentrated in a narrow range f angles (typically within 10-5 radians) with a maximum at the Bragg angle θB defined above.
The Uncertainty Principle
The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements. There is likewise a minimum for the product of the uncertainties of the energy and time.
This is not a statement about the inaccuracy of measurement instruments, or a reflection on the quality of experimental methods; it arises from the wave properties inherent in the quantum mechanical description of nature. Even with perfect instruments and technique, the uncertainty is inherent in the nature of things.
Graphical Interpretation of HUP
Important steps on the way to understanding the uncertainty principle are wave-particle duality and the De Broglie hypothesis. As you proceed downward in size to atomic dimensions, it is no longer valid to consider a particle like a hard sphere, because the smaller the dimension, the more wave-like it becomes. It no longer makes sense to say that you have precisely determined both the position and momentum of such a particle. When you say that the electron acts as a wave, then the wave is the quantum mechanical wave function and it is therefore related to the probability of finding the electron at any point in space. A perfect sine wave for the electron wave spreads that probability throughout all of space, and the "position" of the electron is completely uncertain.
Why do we need quantum mechanics?
Classical mechanics does not provide an accurate description of matter on the scale of atoms and molecules. Electrons around a nucleus or nuclei do not behave like planets orbiting the sun.
Experiments show that when observing the properties of very small bits of matter, such as a single electron, the matter exhibits wave-like properties. Quantum mechanics is the mathematical description of matter on the atomic scale.
When moving particles hit a gap in a barrier they will generally trickle out in a thin stream, but waves will spread out in every direction to form a circle. This is called diffraction.
If there are two gaps in the
barrier, the two waves will interfere to create an interference pattern. The pattern is formed by alternating bands of wave reinforcement and cancellation. According to classical laws, particles do not create an interference pattern. This setup is called adouble-slit experiment. For all these reasons, at the beginning of the 20th century everybody thought particles and waves were totally separate phenomena. But that view had to change.
When Einstein introduced the concept of photons, he showed that waves could have particle aspects. 18 years later Prince Louis de Broglie showed that every particle had wave aspects. Combining the energy-frequency relation for photons with Einstein's famous equation E=mc2, where "E" is energy, "m" is mass and "c" is the speed of light, de Broglie argued that every particle had a frequency, something only waves have.
This was hard to explain since particles are supposed to have an exact location, whereas waves are spread over a range of locations. But remember that two waves can interfere to form another, more complex wave. If we add enough waves together, we can make a wave packet with a range of locations that's as small as we want. This doesn't give a particle an exact location, but it looks like an exact location and so explains why we see it that way. The trick is that whenever we measure the position of the particle, the range of locations narrows just enough so that we never see the wave, a process called wave collapse. The reason we know the wave exists is because it determines the possible range over which we might see the particle. If this seems farfetched to you, you're in good company: what exactly happens when a particle-wave collapses is still debated by physicists today. But it works, so let's put off discussing wave collapse for a bit.
Recall the double-slit experiment from above, but this time let's do it with electrons, which are one of the fundamental particles. On the other side of the barrier we put a screen that emits light when an electron hits it. Now, the electron hitting the screen amounts to a measurement of its position, so the electron-wave collapses and we only see a tiny speck. But repeating this process many times creates an interference pattern! The electron-wave is determining the possible range of locations: an electron is very likely to appear at points of reinforcement, and has zero chance of appearing at points of cancellation. Uncertainty:
Since a particle is a wave, it simultaneously exists over a range of locations. This range is called the uncertainty of the particle's position. In the picture to the right, the triangle in front of the "x" is the Greek letter delta. If "x" stands for the particle's position, then "delta x" represents the range of possible locations for the particle, or the uncertainty of "x".
The narrower you want to make the range of positions, the more waves you have to add together. This creates a wider range of wavelengths for the wave packet. According to de Broglie, a particle's momentum (its speed times its mass) is related to its wavelength. Therefore a narrow range, or small uncertainty, in position means a wide range, or large uncertainty, in momentum.
Werner Heisenberg derived these uncertainty relations, between position and momentum, and between energy and time. The "h" is a number called "Planks constant" (the line through it means divide by twice the number "Pi"). They put an absolute limit on how much we can know about a particle.
Wave Functions and Orbitals
We still talk about the Bohr model of the atom even if the only thing this model can do is explain the spectrum of the hydrogen atom because it was the last model of the atom for which a simple physical picture can be constructed. It is easy to imagine an atom that consists of solid electrons revolving around the nucleus in circular orbits.
Erwin Schrödinger combined the equations for the behavior of waves with the de Broglie equation to generate a mathematical model for the distribution of electrons in an atom. The advantage of this model is that it consists of mathematical equations known as wave functions that satisfy the requirements placed on the behavior of electrons. The disadvantage is that it is difficult to imagine a physical model of electrons as waves.
We know that the de Broglie wave of a particle exists because particles can form interference patterns. But we can never see the wave because whenever we measure its position the wave collapses. The most common interpretation is that the particle doesn't have a definite position until we measure it.
The Schrödinger model assumes that the electron is a wave and tries to describe the regions in space, or orbitals, where electrons are most likely to be found. Instead of trying to tell us where the electron is at any time, the Schrödinger model describes the probability that an electron can be found in a given region of space at a given time. This model no longer tells us where the electron is; it only tells us where it might be.
Terminology
operator - a series of mathematical steps.
Hamiltonian, - the operator that describes the energy of an electronic system.
wave function, - a mathematical function that describes a wave-like shape.
eigen value - a value obtained from operating on a wave function.
The behavior of a quantum particle is completely described by the Schrödinger equation:
A given Hamiltonian operator will have a series of wave function that satisfy the Schrödinger equation. These wave functions are called eigen functions. The Hamiltonian operating on the eigen functions produces the eigen values, E, which are the allowed energies of the system.
Particle Confinement
The uncertainty principle contains implications about the energy that would be required to contain a particle within a given volume. The energy required to contain particles comes from the fundamental forces, and in particular the electromagnetic force provides the attraction necessary to contain electrons within the atom, and the strong nuclear force provides the attraction necessary to contain particles within the nucleus. But Planck's constant, appearing in the uncertainty principle, determines the size of the confinement that can be produced by these forces. Another way of saying it is that the strengths of the nuclear and electromagnetic forces along with the constraint embodied in the value of Planck's constant determine the scales of the atom and the nucleus.
The following very approximate calculation serves to give an order of magnitude for the energies required to contain particles.
Particle in a Box
The idealized situation of a particle in a box with infinitely high walls is an application of the Schrodinger equation which yields some insights into particle confinement. The wave function must be zero at the walls and the solution for the wave function yields just sine waves.
The longest wavelength is
and the higher modes have wavelengths given by
When the momentum expression for the particle in a box :
is used to calculate the energy associated with the particle
Though oversimplified, this indicates some important things about bound states for particles: 1. The energies are quantized and can be characterized by a quantum number n 2. The energy cannot be exactly zero. 3. The smaller the confinement, the larger the energy required.
If a particle is confined into a rectangular volume, the same kind of process can be applied to a three-dimensional "particle in a box", and the same kind of energy contribution is made from each dimension. The energies for a three-dimensional box are
This gives a more physically realistic expression for the available energies for contained particles. This expression is used in determining the density of possible energy states for electrons in solids.
Particle in a Finite-Walled Box
A simple case to illustrate quantum mechanics is to consider a particle in a one-dimensional box. (A better description might be a bead on a string, but particle-in-a-box is the accepted name.)
The electron moves between two walls at x=0 and x=L. The potential energy of the particle is zero between the walls and infinity at the walls. What this statement is saying is that the particle can only be between x=0 and x=L.
V= V=0 V=
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0 L
x -->
By classical mechanics we could roll a particle along the x-axis in this box and predict its position at any later time. That is not true for a particle in the quantum regime. The behavior of a particle is completely specified quantum mechanically by the Schrödinger equation:
Where is the Hamiltonian that describes the system, and and E are eigen functions and eigen values, respectively, that satisfy this equation.
For any particle, the total energy of that particle is the sum of the kinetic and potential energy. For the one-dimensional case is:
Thus, for a particle in one dimension (x-axis) the Schrödinger equation is:
where: h is Planck's constant, m is the mass of the particle (an electron in this case), and V is the potential energy. Since V = 0 in the box, this equation simplifies to:
The general solution for this type of differential equation is:
We can use the boundary conditions and physical reality to narrow down the solution. Since = 0 at x = 0, the cos term is not reasonable and we get:
k must be an integral multiple of , thus, the allowed wave function are:
n is an integer greater than zero we call a quantum number. (n = 0 no particle in the box)
The electron must be somewhere in the box so normalize to find A.
(This postulate is made in analogy to classical EM theory: I ~ A2, light intensity is proportional to the amplitude of the electric field wave squared.)
We now have our allowed wave function:
Substituting n into the Schrödinger equation gives the allowed energies:
Look at this result: Wave properties and boundary conditions lead to quantized energies.
Note that as the size of box changes the energies change proportional to 1/L2. As the box shrinks, the possible energies of the particle increases (see the example below).
For large quantum numbers (needed for reasonable energies at macroscopic dimensions) the spacing between nodes in the wave function become too small to measure. The result is that the particle has an equal probability of being anywhere in the box. Similarly, the relative difference in energy between levels large becomes too small to measure, so the energy the particle can have appears continuous rather than quantized when n becomes large. Both position and energy can be described by classical mechanics at large n. This result is called the Bohr Correspondence Principle.
For the finite potential well, the solution to the Schrodinger equation gives a wave function with an exponentially decaying penetration into the classically forbidden region.
Confining a particle to a smaller space requires larger confinement energy. Since the wave function penetration effectively "enlarges the box", the finite well energy levels are lower than those for the infinite well. Varieties of Wave Equations
Early in the 20th century, electrons were shown to have wave properties, and the wave-particle duality became a part of our understanding of nature. The mathematics for describing the behavior of such electron waves might be expected to be similar to that for describing classical waves, such as the wave on a stretched string
or a plane electromagnetic wave
The wave equation developed by Erwin Schrodinger in 1926 shows some similarities in its one-dimensional form:
The Postulates of Quantum Mechanics
1. Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system. 2. With every physical observable q there is associated an operator Q, which when operating upon the wave function associated with a definite value of that observable will yield that value times the wave function. 3. Any operator Q associated with a physically measurable property q will be Hermitian. 4. The set of Eigen functions of operator Q will form a complete set of linearly independent functions. 5. For a system described by a given wave function, the expectation value of any property q can be found by performing the expectation value integral with respect to that wave function. 6. The time evolution of the wave function is given by the time dependent Schrodinger equation. The Wave function Postulate
It is one of the postulates of quantum mechanics that for a physical system consisting of a particle there is an associated wave function. This wave function determines everything that can be known about the system. The wave function is assumed here to be a single-valued function of position and time, since that is sufficient to guarantee an unambiguous value of probability of finding the particle at a particular position and time. The wave function may be a complex function, since it is its product with its complex conjugate which specifies the real physical probability of finding the particle in a particular state.
Constraints on Wave function
In order to represent a physically observable system, the wave function must satisfy certain constraints:
Further discussion 1. Must be a solution of the Schrodinger equation.
2. Must be normalizable. This implies that the wave function approaches zero as x approaches infinity.
3. Must be a continuous function of x.
4. The slope of the function in x must be continuous.
Specifically
must be continuous.
These constraints are applied to the boundary conditions on the solutions, and in the process help determine the energy eigenvalues.
Probability in Quantum Mechanics
The wave function represents the probability amplitude for finding a particle at a given point in space at a given time. The actual probability of finding the particle is given by the product of the wave function with its complex conjugate (like the square of the amplitude for a complex function).
Since the probability must be = 1 for finding the particle somewhere, the wave function must be normalized. That is, the sum of the probabilities for all of space must be equal to one. This is expressed by the integral
Part of a working solution to the Schrodinger equation is the normalization of the solution to obtain the physically applicable probability amplitudes.
Normalization Examples
In order to use the wave function calculated from the Schrodinger equation to determine the value of any physical observable, it must be normalized so that the probability integrated over all space is equal to one.
Fundamental Forces
Operators in Quantum Mechanics
Associated with each measurable parameter in a physical system is a quantum mechanical operator. Such operators arise because in quantum mechanics you are describing nature with waves (the wave function) rather than with discrete particles whose motion and dynamics can be described with the deterministic equations of Newtonian physics. Part of the development of quantum mechanics is the establishment of the operators associated with the parameters needed to describe the system. Some of those operators are listed below.
It is part of the basic structure of quantum mechanics that functions of position are unchanged in the Schrodinger equation, while momenta take the form of spatial derivatives. The Hamiltonian operator contains both time and space derivatives.
The Operator Postulate
With every physical observable there is associated a mathematical operator which is used in conjunction with the wave function. Suppose the wave function associated with a definite quantized value (eigenvalue) of the observable is denoted by n (an eigenfunction) and the operator is denoted by Q. The action of the operator is given by
The mathematical operator Q extracts the observable value qn by operating upon the wave function which represents that particular state of the system. This process has implications about the nature of measurement in a quantum mechanical system. Any wave function for the system can be represented as a linear combination of the eigenfunctions n ( see basis set postulate), so the operator Q can be used to extract a linear combination of eigenvalues multiplied by coefficients related to the probability of their being observed (see expectation value postulate).
Wave function Contexts
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