Maths and Physics Tuition/Tests/Notes

Some Strange Implications of Quantum Mechanics

Quantum m,echanics can have some very strange implications. A particle may be in several places at one. This is a consequence of the state of the particle being described by a wavefunction, with the probability of a particle being in a region of space...

The Connectin Between Kinetic Energy in Newtonian and Relativistic Mechanics

In Newtonian mechanics, the momentum and kinetic energy are given by and The relationship between momentum p and kinetic energy is often written as Many equations and concepts in Newtonian mechanics have close analogies in special relativity. For...

Quantum Mechanical Operators

and dymamics 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...

Mechanical Determinism

Newtonian mechanics is a complete system. Given a state of a system of particles and the mass, velocity and position of each particle, the state of the system – the position and velocity of each particle can be determined at any subsequent or previous...

The Mechanics of Atomic Spectra

In an atom, electrons occupy orbits, each with a particular energy. The orbits and energies are specific to each element. This energy is negative because the electron and nucleus are oppositely charged, so attract each other and energy must be given to...

The Mechanism of Lightning Production

Static electricity is commonly produced when insulating materials rub together. When fur is rubbed against an ebonite rod, the fur becomes negatively charged and the rod becomes positively charged. Electrons are transferred from one material to the rod...

The Mechanism of Light Production by a Gas Discharge Tube

Gas is normally not a good conductor of electricity. The molecules of the gas are electrically neutral and the gas contains very few ions. This means that in order for a gas to conduct electricity, a source of electrons is needed. In a gas discharge...

Principle of Least Action

principle that can be used to obtain the equations of motion for that system in a way that is independent of Newtonian mechanics. The action principle is preceded by earlier ideas in surveying and optics. The rope stretchers of ancient Egypt stretched...

Postulates

to 1. Thewavefunction must be single-valued, continuous, and finite. Postulate 2. To every observable in classical mechanics therecorresponds a linear, Hermitian operator in quantum mechanics.If we require that the expectation value of an operator...

The Born Interpretation

The fundamental equation of quantum mechanics is the Schroedinger equation. It was introduced as a wave equation which did not contradict the relation between the energy and the momentum of a particle in classical mechanics. As a result, the wave...

The Einstein Rosen Podolsky Experiment

other. We have a source that emits electron-positron pairs, with the electron sent to destination A. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. This can be viewed...

Angular Momentum Operators and Commutation Relations in Quantum Physics

In is one of the fundamental differences of quantum physics with classical mechanics that in the quantum world we cannot know all measurements to absolute precision. Some quantities 'pair up' so absolute knowledge of one precludes absolute knowledge of...

The Schrodinger Equation

The general equation expressing the conservation of energy of a particle in classical mechanics is In quantum mechanics we have something very similar, with the intoduction of a concept called the wavefunction. or equivalently The first of these is...

Definitions

Terms commonly used everyday by ordinary people may have a slightly different meaning in physics. Many people for example confuse velocity and speed. These are not the same, the difference being that velocity is the speed of a body in a certain...

Introduction to Angle Action Variables

taken together are called angle action variables. Angle action variables are of fundamental importance in quantum mechanics, where they are used for example to model the radiation emitted from quantum systems. Most Hamiltonians for functions of both the...

The Cauchy - Riemann Equations

of the velocity field of a steady incompressible, irrotational flow in two dimensions. The equations are occur in fluid mechanics from consideration of the velocity potential and are found from the velocity potential The condition that the flow be...

Angular Momentum Operators

momentum is so important because it is conserved in any isolated system. The same relationship, (1) holds in quantum mechanics, with and being the position and momentum operators respectively: can be evaluated as the determinant of a matrix, the first...

Expectation Values

value between certain limits the we evaluate an integral. In this case where the range of possible values is In quantum mechanics we have something similar. Remember that the probability density of a particle is given by where is the operator of the...

Quantum Tunnelling

0 in region III there is a non zero probability of finding the particle in region III, which is forbidden by classical mechanics since it has insufficient energy to surmount the barrier.

Reflection From a Potential Barrier

it is not automatically relected, even if it has insufficient energy to over come the barrier according to classical mechanics. If the potential barrier has finite height, the wavefunction partially penetrates the barrier. If the barrier has finite...

Bohr's Model of the Hydrogen Atom

of angular momentum in units of where This means we can write We can use the ordinary rules of classical Newtonian mechanics to derive the equation giving the differences in the energy levels of the electrons in the Hydrogen atom. We can equate the...

The Paul Exclusion Principle

possess an intrinsic angular momentum whose value is times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics, fermions are described by "antisymmetric states" such that if any two are interchanged, a phase change in the...

Acceleration Time Graphs

Acceleration time graphs indicate the acceleration relative to particular point as a vector. One direction is taken as positive and one direction is taken as negative, so only these this forwards and backwards motion can be displayed. At any time t the...

Centres of Gravity

Every object has a balance point. You can support it on a finger, it being not too heavy, and if, with you finger positioned in a certain place, the object will balance. Your finger will exert an upwards force on the object to stop it falling. Opposing...

Coefficient of Friction

When a block resting on a surface is pushed (or pulled) with a force P, it experiences a force of friction, Fr, that opposes the movement of the block. If the force pushing the block increases, so does the frictional force, but only up to a limit. The...

Components of Projectile Motion

If a ball is thrown up in the air, not vertically and not horizontally but at an angle to the horizontal, then it follows a path of the form Any path of this form is called a parabola. The ball, a projectile, describes a parabola, and this is true for...

Different Types of Force

There are many different type of force and some of them have the same origin – the electric force. The following table provides a summary. Name of force Description Gravitational The force between objects of mass and separated by a distance as a result...

Displacement Time Graphs

Displacement time graphs indicate the displacement relative to particular point as a vector. One direction is taken as positive and one direction is taken as negative, so only these this forwards and backwards motion can be displayed. At any time t the...

Dynamic and Static Friction

Friction opposes the relative motion of two surfaces in contact. It occurs because surfaces are not smooth on the microscopic scale. If the surfaces are not moving relative too each other, then there is static friction. If they are in relative motion,...

Efficiency and Power

Not all energy does something useful. When work is done on an object, usually some energy is used to overcome a friction or air resistance force. This energy becomes internal energy of the molecules of a material. This energy takes the form of a random...

Elastic and Inelastic Collisions

Momentum is always conserved in collisions. Energy however is not always conserved. In fact collisions can be classified into three types, according to whether kinetic energy is conserved or not: Elastic Collision – Kinetic Energy is conserved. In...

Energy

When work is done on an object, the object gains energy and the medium that does work loses energy. The amount of energy transferred is equal to the work done. It is not always equal to the amount of energy transferred to the object. Some energy is...

Energy of an Orbiting Satellite

The gravitational potential energy of an orbiting satellite of mass m is given by (1) where is the mass of the Earth. We can find an expression for the kinetic energy by considering the equation for a satellite in a circular orbit. For such a...

The Equations of Uniform Motion

If the acceleration is constant and the motion is in a straight line then simplifications become possible. We can define five variables: – displacement – initial velocity – final velocity – acceleration – time and write down five equations that link...

Escape Speed or Velocity

If you throw a ball up it the air, it goes up. The faster you throw it the higher it gets. As the ball rises, the ball slows and the pull of the Earth's gravity gets weaker. At an infinite distance, the pull of the Earth's gravity will be zero, but you...

Falling Objects

A very important example of uniformly accelerated motion, as long as we ignore air resistance, is the vertical motion of an object in a uniform gravitational field, called free fall. Taking down as positive, we can draw graphs of acceleration, velocity...

Frames of Reference

If two objects are moving along the same line but at different speeds, then they will have a relative velocity relative to one another. We can find this by simple addition or subtraction. The relative speed of car A relative to car B above is 40-30=+10...

Free Body Diagrams

A free body diagram illustrates all the forces acting on a particular object and leaves us in a position to apply Newtons second law – – easily. Often the forces will be balanced so the forces vertically and horizontal are equal, Generally this will...

Geostationary or Geosynchronous Orbit

The closer a satellite is to the Earth, the faster it moves. For a certain radius of orbit the satellite will move at such a speed that it is always over the same spot over the Earth's surface – actually above the equator. Such an orbit is called a...

Gravitational Potential

We can define a quantity that gives a measure of the amount of energy a body of unit mass has when in a gravitational field. This quantity is called the gravitational potential and is defined as where is the work that must be done to remove a body to a...

Inertial Mass, Gravitational Mass and Weight

We mislead ourselves when we say we are measuring the mass of an object. If fact we are measuring the force of gravity on that object and changing that force into a mass using a calibrated scale. If we were to 'measure' the mass of an object at...

Instantaneous Average

Speed is not merely distance divided by time. Intuitively we mean the distance travelled in a particular time divided by the time taken to travel that distance. The calculated speed will be the average speed over that time interval, but the speed may...

Kepler's Third Law

Kepler's laws of motion describe the orbits of planets around the Sun. They were were a bridge between the Aristotlean view of the Solar System, which described in error and did not explain, and the Newtonian view, which described (almost) correctly...

Momentum

Linear Momentum – units kg m/s - is defined as the product of mass and velocity. It is a vector, and is of fundamental importance because in any collision or any isolated system it is conserved as a consequence of Newton's third law. We can write this...

Newton's Second Law of Motion

Newtons first law of motion states that a body subject to no net external forces continues in a state of uniform motion. His second law of motion in a sense makes his first law complete by telling us what will happen in the event of a body being...

Newton's Third Law of Motion

Newton's third law states: To every action there is an equal and opposite reaction. Newton's third law implies that forces always come in pairs. In fact, these force pairs have the same origin and nature. If one is gravitational, then so is the other;...

Some Basics of Forces

Forces occur whenever the direction of motion of a body changes – it accelerates - or whenever an object changes shape. When a football is kicked for example, the ball may change shape slightly – temporarily – and will change direction – often towards...

The Approximate Expression for the Gravitational Potential Energy

The equation (increase in) Gravitational Potential Energy= is only an approximation when is is small compared to the radius of the Earth. The graph of gravitational potential against the distance from the Earth's centre is shown below. To move from a...

Torque and Moments

The motion of a rigid body is in general a combination of translation and rotation. The two tyopes of motion can be treated independently. A translation takes place when every particle in a rigid body has the same velocity while a rotation is when...

Uniform Circular Motion

The phrase 'uniform circular motion' describes planar motion in a circle at constant speed. If the motion takes place on the Earth, this almost certainly means that the motion is horizontal. For example a mass suspended at the end of a string may be...

Uniformly Accelerated Motion

If the acceleration of a body is uniform or constant, simplifications become possible so it is important to determine if the acceleration is uniform to a good approximation. Possible methods of finding if the acceleration of a body is uniform include:...

Velocity Time Graphs

Velocity times graphs indicate the velocity as a function of time. One direction is taken as positive and one direction is taken as negative, so only this forwards and backwards motion can be displayed. At any time t the velocity can be read of the...

Weight

Weight and mass are often taken to be synonymous. In fact they are very different concepts, but people are easily confused because although weight is actually a force measured in newtons and mass is the amount of matter measured in kg, people often say...

Weightlessness

One way of defining the weight of an object is to say that the weight is the value of the force recorded on a supporting scale. If the scales were set up in a lift, the value they record would depend on the magnitude an direction of the acceleration of...

Work Done by a Force

A force is done whenever the point of application of the force moves in the direction of the force. If a force pushes a block then the block does work. The work done by the force however is not in general to the force multiplied by the distance. Only...

Canonical Transformations

Coordinate transformations or changes of variables are useful because a suitable choice of coordinates or variables can dramatically simplify a problem. For example cannot be evaluated by inspection but on using the integral becomes on using the...

An Example of a Time Dependent Hamiltonian and Lagrangian

Deriving the Hamiltonian and Lagrangian for a time dependent system is not much more complicated than for the time independent case. The pendulum of mass and length below is made to oscillate at A with the distance OA given by The potential energy is...

Equations of Curcular Motion

For motion in a straight line with constant acceleration, we can use the SUVAT equations of motion: These have exact equivalents for motion in a circle with constant angular acceleration. In these equations, and We can derive the circular equations of...

Finding a Generating Function From a Given Canonical Transformation

Generating functions have many uses and it is useful to be able to construct one from a given transformation. We can do this from the defining relationship between the generating function and the transformation. Example Show that the transformation is...

Generalized Coordinates

In order to describe the motion of a system mathematically we need to be able to specify the instantaneous configuration of the system - the position. For example the motion of a projectile can be specified by the horizontal distance of the projectile...

Generating Functions for Transformations of Coordinates

In order to describe the motion of a system mathematically we need to be able to specify the instantaneous configuration of the system - the position. For example the motion of a projectile can be specified by the horizontal distance of the projectile...

Infinitesimal Canonical Transformations

Perturbed and unperturbed phase curves of a Hamiltonian system can be connected via a canonical transformation. This suggest that perturbation theory for Hamiltonian systems is really the study of canonical transformation which depend upon a parameter...

Introduction to Perturbation Theory

Real problems rarely have Hamiltonians with equations of motion having simple solutions depending upon elementary functions, so some method of approximating the solutions is needed. Typically we start with a Hamiltonian of the form (1) where the...

Making Equations Dimentionless

It is often useful to remove the units from an equation. We can see which physical mechanisms are more important and the equation to be solved is simpler. We scale all variables so that the variables become dimensionless. We can solve the equation...

Moments of Inertia of Some Basic Shapes

Possible Forms of Generating Function

Given an arbitrary canonical transformation it may not be possible to treat and as independent variables since the condition for this to be possible is that the equation can be solved to give in terms of and so so this transformation cannot be applied...

Preservation of Area by Time Independent Systems

A system influenced by time dependent forces or which is represented in a rotating or non inertial reference frame has a Hamiltonian which depends explicitly on time, The rate of change of the Hamiltonian is given by On using Hamilton's equations of...

Proof of Conservation of Angular Momentum

Theorem Angular momentum is conserved for a body subject to no net external torque. Proof Let a body consist of particles at position vectors The motion of each particle is determined by Newton's second law or Taking the cross product with and summing...

Proof of Newton's First Law of Motion

Theorem A body continues in a state of uniform motion unless acted on by an external force. Proof Let a body consist of particles of mass at position vectors The equation of motion for particle is The force acting on the body is The position vector of...

Proof of the Law of Conservation of Momentum

It is not obvious why momentum should be conserved. Momentum is avery abstract concept, and the principle of conservation of momentummay be seen as a consequence of other, fundamental Laws of physics: The principle of Relativity The Principle of...

Proof That the Moment of a Force Acting on a Body Equals Rate of Change of Angular Momentum of the Body

It is not obvious why momentum should be conserved. Momentum is avery abstract concept, and the principle of conservation of momentummay be seen as a consequence of other, fundamental Laws of physics: The principle of Relativity The Principle of...

Proof That Trajectory of BodyAbout Another Body is a Conic Section

Suppose a body moves under the influence of a single mass producing a gravitational field. The equation of motion of the body is Taking the cross product of this equation with gives Now since angular mometum is conserved. Hence a constant vector. We...

Condition for a Transformation to Preserve Hamiltonian Form

The area of a region of the phase space diagram at time is given in terms of the area at some earlier time by Differentiating this expression gives or is a solution curve of the system so then for a transformation The second bracket equals 0 for...

The Hamiltonian

The Hamiltonian represents the energy of the system which is the sum of kinetic and potential energy, labelled and respectively. For a one dimensional system, we may write so where Note that is a function of only and is a function of only. In general...

The Lagrangian Equation of Motion

The Lagrangian is defined as where and is a function of Hence The first of Hamilton's equations gives so the bracketed term vanishes and leaves The other of Hamilton's equations is and we can use to give This is Lagrange's equation of motion. It is a...

The Relationships Between Different Generating Functions

There may be more than one generating function for the same transformation, so it seems logical that relationships exist between each generating functions. These do in fact exist. Example: Find the relationship between and Hence The set of...

The Rocket Equation

Without gravity With Gravity:

Time Dependent Canonical Transformations

Time dependent canonical transformations are very similar to the time independent case, and as with the time independent case, the preservation of area is a part of the analysis. Consider the transformation The condition for such a transformation to...

Damped and Forced Vibrations

Free undamped, unforced vibrations in simple harmonic motion obey the equation or Typically the vibration will be subject to a resistive term R, which for low speeds is proportional to velocity and in the opposite direction hence The equation for...

Alternative Forms of The Continuity Equation

The continuity equation is usually written (1) where is the density of the fluid at a point. < > It may also be written (2) or (3) where is the Stokes derivative. < > (2) follows from (1) on using the identity < > To derive (3) write

Bernoulli's Equation for a Steadily Flowing Liquid

The Navier - Stokes equation is - a complicated equation which is difficult to solve, except in situation allowing much simplification. If we make the following assumptions, many of important features of the flow are retained. The fluid is inviscid, so...

Classificatin of Open Channel Flow

Consider the steady, uniform flow of an inviscid, incompressible liquid in an open channel with a rectangular cross section of constant width. If the width is and the depth is the volume flow rate where is the speed of the liquid. is constant from the...

Deep Water Gravity Waves - Finite Depth

The equations satisfied by waves are for water of depth (1) at (2) at (3) where is the velocity potential. (1) can be solved by separation of variables technique. Assume (there will also be an arbitrary factor of which we deal with later). (1) becomes...

Deep Water Gravity Waves - Infinite Depth

The equations satisfied by waves are for water of depth (1) at (2) at (3) (1) can be solved by separation of variables technique. Assume (there will also be an arbitrary factor of which we deal with later). (1) becomes since the left hand side is a...

Deriving the Equations of Water Waves

Water waves obey simple differential equations derived using simplifying assumptions of incompressibility and irrotationality. If the flow is irrotational we can define a velocity potential satisfying If the fluid is incompressible then for waves...

Equation of Motion For a Viscous Fluid

The eqution of motion for an inviscid fluid is < > where is the density, is the pressure and is the potential as a function of position. < > To include viscosity effects we can just add a viscosity term to the right hand side. < > The equation becomes...

Equation if Motion For an Inviscid, Incompressible Fluid

For an incompressible fluid, so If the body force per unit volume is applying Newton's Second Law to a cube of unit volume to obtain We can also write where is the pressure in the fluid. Equating these gives (1) Consider the motion of a particle in the...

Incompressible Fluids

If a fluid is incompressible and is the velocity vector field that describes the velocity of the fluid at each point then This equation follows from the continuity equation First write the continuity equation as Since the fluid is incompressinble There...

Kelvin's Theorem

Kelvin's circulation theorem states: In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant. where is the circulation around a material contour which may vary with...

Newtonian Fluids

Viscosity may be thought of as a fluids resistance to the shearing forces acting on the fluid. We may picture a fluid between two plates, one stationary and one moving parallel to the first. The relative motion of the two plates cause shearing forces...

Pathlines

Pathlines in fluids are the paths taken by the individual particles of the fluid as they travel from point to point. We can find the pathlines taken by the particles of a fluid if we know the velocity of the particles of the fluid. In two dimensions,...

Paths of Particles in Water Waves

The velocity potential for water waves in water of finite depth, h, can be written From this we obtain and Integrating these equations with respect to gives and then This is the equation of an ellipse. For large and close to 0, far from the bottom...

Proof That Loss of Fluid Per Unit Volume Equals Divergence of Velocity Field

The x component of the velocity at the centre of the face ABCD above is The x component of the velocity at the centre of the face EFGH above is The volume of fluid crossing ABCD per unit time is The volume of fluid crossing EFGH per unit time is The...

Streamlines

The field lines of the velocity vector field at a particular instant of time are called streamlines. The streamlines are visualised by taking photographs of the fluid. If the velocity field changes with time, then the streamlines will change also, but...

The Navier - Stokes Equation

The Navier – Stokes equation models the behaviour of a fluid element because of the forces acting on it, including viscous forces. When the viscous forces are ignored, the equations become Euler's equation. The equation is difficult to solve, and...

The Reynolds Number

The Reynolds number Re is a dimensionless number, the ratio of inertial forces to viscous forces i.e. where is the mean relative velocity between object and fluid (m/s). is a characteristic linear dimension (m). is the dynamic viscosity of the fluid...

Total Force Acting on a Cube of Fluid

For a incompressible fluid at rest there are no shear or deforming forces and Any stresses are normal to any surface drawn in the fluid and the pressure inside the fluid is the same in all directions. Since though the fluid is incompressible the...

Velocity Distribution of Fluid Between Rotating Pipes

Consider a fluid of viscosity between concentric rotating pipes of radius with rotating with angular velocities respectively. For two dimensional flow in the z – direction the shear stress where is the shear strain and is the shear stress. If the flow...

Velocity Potential

Velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region – no sources or sinks - and is irrotational, so the velocity field has zero curl: As a result, can be represented as the gradient of a scalar function: in...

Vorticity

Vorticity, labelled is the tendency of fluid to spin. It may vary from point to point in a fluid, and with time. It arises because different parts of the fluid are in relative motion, so any fluid element between these two parts exhibits a tendency to...

Water Waves

Most water waves are formed as a result of changes in the pressure and velocity of air close to the water surface. The largest waves are however formed by tides, currents, earthquakes etc. Increasing windspeed is associated with increasing wave height....

Water Waves on a Sloping Beach

To the landperson waves are most encountered at the beach, or swimming near the beach. Far from the beach the waves are nearly uniform progressive waves with wavelength from a few metres to a few hundred metres and speeds ranging from 4m/s to 20m/s....

Constants

Avagadro's Constant Planck's Constant Mass of Electron Mass of Proton Mass of neutron Elementary Charge Bohr Magneton Bohr Radius Compton Wavelength of Electron Coulomb's Law Constant Electron Charge to Mass Ratio Rydberg Constant for Hydrogen Speed of...

Dispersion Relations

A particle does not occupy a fixed, definite position in space. It is smeared out over a definite none zero volume in the form of a wavepacket. In general the wavepacket is made of of many differentfrequencies or wavelengths. They interfere...

Features of Bound State Wavefunctions

The particle moves fastest in regions of minimum potential energy. Since it moves fastest and has most energy there, it's wavelength is shorter. It spends least time in those regions and so the probability of finding the particle there is low, hence is...

Features of Solutions of the Schrodinger Equation for the Hydrogen Atom

The general solution of the Schrodinger equation corresponding to a principal quantum number is a polynomial in of degree with circles of zero electron density multiplied by a polynomial in ( or or both) of degree with radial lines of zero electron...

Hermitian Operators

Quantum Mechanical operators are hermitian. If a quantum mechanical operator is represented by a matrix then so that is equal to the complex conjugate transpose of For example, the spin Pauli operators may be represented by the matrices Every quantum...

Linear Combinations of Eigenfunctions

An eigenfunction which represents a possible state of a particle is any solution of the Schrodinger equation which may be written in the form where A linear combination of any number of eigenfunctions is also a possible wavefunction. Proof: Hence the...

Normalizing the Wavefunction

The Born interpretation gives the probability of finding a particle with wavefunction - I have shown the wavefunction here to be a function of here, though I need not have done and do not use this below - in the volume of space between and is The...

Probability Current or Probability Flux

The probability density of a particle with wavefunction – or statefunction – is The probability density function changes in space, but it may also change in time. If the probability density is a function of time, then the particle will be moving and...

Properties of the Eigenfunctions

The eigenfunctions are the solutions of the eigenfunction equation the solutions for the one dimensional simple harmonic oscillator case, are polynomials in multiplied by a gaussian If the are normalized to unity they have the following properties: The...

Selection Rules for Transitions of Electrons Between Atomic Energy Levels

In spectral phenomena such as the Zeeman it becomes evident that transitions are not observed between all pairs of energy levels. Some transitions are "forbidden" while others are "allowed" by a set of selection rules. That a...

The Bohr Magnetron

We can picture an electron in an atom moving in a circle of radius with speed The moving electron is equivalent to a current loop. A current loop with area and current has a magnetic dipole moment given by so for the electron above To find the current...

Compton Scattering

Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The elastic scattering – implying conservation of energy - of photons in matter results in a decrease in energy (increase in wavelength)of an X-ray or gamma ray...

The Infinite Square Well

The infinite square well potential is given by: This is illustrated below. A particle under the influence of such a potential is free - no forces act - between and and is confined to that region by the need to have an infinite energy in order to travel...

The Leonard - Jones Potential

The Lennard-Jones or L-J potential is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. The expression of the L-J potential is where is the depth of the potential well, is the (finite) distance...

The Separation of Variables Method - The Simple Harmonic Oscillator

In two dimensions Schrodinger's equation takes the form (1) Because x and y appear in the equation, we must assume is a function of both and If we assume that is a product of a function of with a function of then Substitution of this expression into...

The Stern Gerlach Experiment

An electron, being a particle with an associated magnetic moment, may be pictured as a little spinning top. In the presence of a magnetic field this magnetic moment will align with the field in one of two ways – either spin up or spin down. The...

The Weak Nuclear Force

One of the four fundamental forces, about a million times weaker than the strong force – hence the name - the weak interaction involves the exchange of the intermediate vector bosons, the W and the Z. Since the mass of these particles is on the order...

Summary of Quantum Numbers For Electrons in Atoms

Solution of the Schrodinger Equation for the electron in the hydrogen atom gives rise to four quantum numbers. 1. The principal quantum number n. The allowed values of n are 1, 2, 3, 4, and so on. It may not be zero. This number along with the orbital...

Table of Normalized Spherical Harmonics

The Schrodinger equation for the hydrogen atom takes the form This equation is separable which means that while the solution is a function of three variables, it is a product of three functions, each one of which is a function of only one variable, The...

Table of the Radial Parts of the Wavefunctions for the Hydrogen Atom

The Schrodinger equation for the hydogen atom takes the form This equation is separable which means that while the solution is a function of three variables, it is a product of three functions, each one of which is a function of only one variable, The...

The Simple Harmonic Oscillator

The simple harmonic oscillator models the motion of a wavefunction subject to the potential function illustrated below. The Schrodinger equation becomes The first few solutions, in order of increasing energy, are shown below. In this table and Quantum...

Analysis of a Plane Blown Off Course

Suppose a plane can fly 300km/h in still air. It wishes to fly from a town A to a town B on a bearing of 030 o to a town B. The wind is blowing at 20km/h on a bearing of 300 o . If the pilot aims the plane on to fly to B, he will be blow off course by...

Proof That Polar Moment of inertia Equals Sum of Moments of Inertia Aboput x and y Axes

The polar moment of inertia of a lamina with surface \[S\] about a axis through the origin perpendicular to the \[xy\] plane is given by \[I_P \int_S r^2 \rho (x,y) ddS\] where \[r^2 =x^2 +y^2 \] The moment of inerta of the lamina about the \[x\] and...

Moment of Inertia of a Cube About an Edge

Supppse a cube of side \[a\] of variable density \[\rho = k(x+y+z)\] with one vertex at the origin at edges parallel to the axes is allowed to rotate about an axis. What is the moment of inertia of the cube about that axis? A volume element...

Motion in a Circle With Non Constant Acceleration

Suppose a particle is moving on a circle of radius \[r\] with at any time \[t\] the ant clockwise angle of the particle from the positive \[x\] axis being given by \[\theta = \frac{t}{t+1}\] /> br /> the angular velocity is \[\omega = \frac{d \theta...

Motion Defined Parametrically With Non Constant Acceleration

Suppose a a particle is moving along a curve with Cartesian coordinates given parametrically in terms of time \[t\] by \[x=\sqrt{t^2+1}, \; y= \sqrt{2t}\] then the distance of the particle from the origin is \[r= \sqrt{x^2+y^2}= \sqrt{t^2+1+2t}=t+1\]....

Angular Velocity and Acceleration Along Curve

Suppose a particle moves along the curve \[y^2=x^3\] . If the horizontal velocity is constant and equal to 2 units/s, find the angular velocity \[\omega\] and the angular acceleration \[\alpha\] . The particle is at a point \[\theta = tan^{-1}...

Velocity and Acceleration of Piston Operated By a Flywheel

A piston moved by a rod fixed to the edge of a rotating flywheel will move backwards and forwards as the wheel rotates. If the radius of the flywheel is \[R\] and the length of the rod is \[l\] . and we let the centre of the flywheel be the origin, then...

Proof of Stefan Boltzmann Law

According to Planck's radiation law, the radiation density is \[I= \frac{2 h}{c^2} \int^{\infty}_0 \frac{f^3}{e^{\frac{hf}{kT}} -1} df=\frac{2 h}{c^2} \int^{\infty}_0 \frac{f^3}{1-e^{- \frac{hf}{kT}}} e^{-\frac{hf}{kT}} df\] . Substitute \[\frac{hf}{kT}...

The Catenary

Suppose a length \[L\] of chain is hung from an uneven ceiling. The ends of the chain are fixed to two points some distance apart. What will be the shape of the chain? Let the chain lie in the vertical \[xy\] plane. An small length \[ds\] of chain of...

Particle Moving Along y Axis on Surface of Rotating Ellipse

Suppose an ellipse with one focus at the origin is made to rotate about the origin in the \[xy\] plane. A particle on the surface of the ellipse is made to move on the surface of the ellipse as the ellipse rotates. What will be the velocity and...

Quantum Physics and Pure Numbers

significant figures.. In the case of numbers this is true even for those numbers like \[e\] , and in the case of quantum mechanics, even for those quantities that are the exact solutions too some exact equation, like the metafunction of an electron...

The Old Style Cathode Ray Tube

The cathode ray tube is a very common device used in television sets and computer screens. Inside the tube is a vacuum, so the electrons are free to move and do not hit any gas molecules. Electrons are emitted from a negative cathode at A, which is...

University Maths and Physics Tuition

Level: Fluid Dynamics Approximation Theory Topology Differential Geometry Calculus Statistics Electromagnetism Quantum Mechanics Non Linear Dynamics Mechanics Group Theory Optics Galois Theory Physical Chemistry Abstract Algebra Number Theory Planetary...

Degree Level Physics Tuition Throughout London By Physics Specialists

be asked. We Tutor These Physics Topics Up To Degree Level - And Have Written Our Own Course Notes Astronomy Classical Mechanics Electricity and Magnetism Experimental Physics Fluid Mechanics Light and Optics Non Linear Dynamics Physical Chemistry...

Postulates of Relativity

Newtonian mechanics is based on two principles: Space and time are absolute. The time coordinates given to observations in all inertial frames are related by a constant time shift. In particular, all observers in all inertial frames agree on the time...

The Relationship Between Spin, Orbital and Total Angular Momentum

For an atom in groups 1 or 2, or periods, 2 or 3 of the periodic table, any electron with spin oriented up will be opposed by an electron in the same otherwise identical set of quantum numbers with opposing spin, and any electron with values of angular...

Sankey Diagrams

Sankey diagrams show what happens to the energy used during a chemical or physical process. The amount of energy input or used up beach part of the process is represented by a bar with an arrow indicated the direction of the process, with the thickness...

Joule's Experiment

James Joule demonstrated the link between mechanical work done and temperature change. He conducted a series of experiments showing that doing work resulted in an increase in temperature proportional to the amount of work done, allowing for...

The Cases For and Against Nuclear Power

Before nuclear power there was the nuclear bomb, and the presence of the nuclear bomb casts a shadow over the debate about nuclear power, even between countries. Many countries think that nuclear bombs are being developed under the cover of nuclear...

A* Maths/Physics Tuition Throughout London & Online Tests/Notes

for maths and physics. Maths Tuition for every A Level and International A Level(CIE/Edexcel) module - Core C1/C2/C3/C4, Mechanics M1/M2/M3/M4/M5/M6, Statistics S1/S2/S3/S4/S5/S6, Further Pure FP1/FP2/FP3/FP4 and Decision D1/D2, and physics tuition for...

Heisenberg's Uncertainty Principle and the Loss of Determinism

Heisenberg was a pioneer of quantum mechanics, presenting a new picture of particles represented as vectors operated on by waves. The matrix formulation accounted for many of the properties of atoms. Much more intimately connected with the name of...

Spacetime

The success of the special theory of relativity in explaining many things that Newtonian mechanics could not while still being consistent with Newtonian mechanics at low speeds gives us a great deal of confidence. Any two inertial observers will agree...

The Mass Energy Relationship

energy can be changed into mass. Mass can be thought of as a sort of 'frozen' energy. According to the laws of Newtonian mechanics a constant force acting on a body of mass m produces a constant acceleration. A graph of speed against time will be a...

Fermat's Principle

the Hamiltonian formulation of geometrical optics, which shares much of the mathematical formalism with Hamiltonian mechanics. It can be generalised to describe the paths taken by material particles, giving the method of variation of parameters and the...

Longitudinal and Transverse Mass

In Newtonian mechanics the mass of a particle is constant and can be expressed as the ratio of the force to the acceleration: The force can be written as the rate of change of momentum: so In Newtonian mechanics and is constant so In special...

A Level Maths Notes - FP3

Natural Climate Change

The weather is obviously not the same from day to day. It is actually not the same from age to age. Over long periods of of time the Earth's climate chages. In the past – only a few hundred years ago – Europe had a 'little ice age', with the Thames...

Convection

Convection is the process of heat transfer by bulk movement of matter. Convection can only take place in a liquid or gas – fluids, since only in fluids are atoms and molecules free to move. When a fluid is heated it expands. If the heating is uneven...

Electric Wind

When a thin wire is connected to a high voltage source, the electric field at the end of the wire can become intense. A wind of charged ions streams away from the end of the wire. This may be demonstrated by placing a candle flame near the end of the...

Joule's Experiments

Joules experiments on work and heat transfer demonstrate that mechanical or electrical energy can be changed into internal energy to produce a rise in temperature. The apparatus for one of his experiments is shown below. Two heavy lead weights are...

Librational Periods

Libration takes place when a body on periodic motion does not have enough energy to complete a full rotation. When it reaches the end points the sign of changes. Librational motion is illustrated in the phase diagram below, the librations taking place...

Second Order Autonomous Systems

Almost all Newtonian mechanical systems are second order or higher. For a second order system, two variables are needed to model the system and locate a given point in phase space. In Cartesian coordinates the equations of motion take the form and...

Simple Harmonic Oscillator With a Resistive Term rv

In general a particle in motion is subject to many forces. There is a frictional or resistive force in most mechanical systems. Typically the resistive force depends on the speed of the vibration, and is always opposed to motion. The simple harmonic...

The Need For Complex Numbers

said to be 'complex' and labelled with the letter : and Complex numbers have many uses, in geometry, electricity, quantum mechanics, trigonometry, number theory... Example: If then and

Thomson's Plum Pudding Model of the Atom

working model. The pudding had a density equal to the overall density of the material, and fitted well into Newtonian mechanics which pictured atoms as billiard balls. It was also not inconsistent with phenomena such as the photoelectric effect – energy...

The Nature of Light

motion to each other. This lead in the late nineteenth and early twentieth centuries to the superceding of Newtonian mechanics first by the special theory of relativity, then by the general theory of relativity.

Intervals Between Events

In Newtonian mechanics, the distance between simultaneous events events is a number that can be agreed on by all observers, and is equal to In special relativity, observers do not agree on the distance or time intervals between events – in particular,...

The Force Transformation

In classical Newtonian mechanics force is defined as rate of change of momentum: Special relativity changes the definition of both momentum and time, so the definition of force must change. If we have inertial frames O and O', with O' moving along the...

Group Actions

extension acts on the bigger field The additive group of the real numbers acts on the phase space of systems in classical mechanics (and in more general dynamical systems): if and is in the phase space, then describes a state of the system, and is...

The Exponential Fourier Transform

non periodic functions and being more suited to solving partial differential equations, and is more applicable in quantum mechanics where it allows the wavepackets representing particles, which are necessarily localized, to be decomposed into a range of...

Table of Properties of Alpha, Beta and Gamma Radiation

The three main types of nuclear radiation are alpha, beta and gamma radiation. Their properties are summarised in the following table. Type Mass in units of atomic mass units Charge Description Alpha 4 2 Consists of 2 protons particles and 2 neutrons...

Making Ice By Evaporation

When a liquid evaporates, the molecules with the most energy evaporates first. The molecules with less energy, are left behind. Having less energy, the are at a lower temperature, since energy – specifically energy of random movement – is a measure of...

The Simple Camera

The camera consists of a lens to focus the image and a film on which the picture is taken. The image is real, inverted, and smaller than the object. A real camera is a lot more complicated, with maybe several lenses, a shutter, a lot of electronics and...

The Wheel and Axis Principle

The wheel and axis principle is used to produce a large turning force. It is most frequently used in the gearboxes of cars. A large force is required to start the car moving (low gear), which can be reduced once the car starts to move (high gear)....

The Chernobyl Disaster

The Chernobly nuclear power station, in Ukraine, was the site of possibly the worst nuclear accident ever. The day before the disaster, plant operators were preparing for ashutdown to perform routine maintenance on one of the four reactors. As part of...

JJ Thomson

JJ Thomson discovered the electron and determined it's charge to mass ratio Experiment showed that if the cathode rays were made to hit a gold leaf electroscope, the leaf would deflect. Investigation implied the cathode rays carried a negative electric...

The Suns Corona

TheSuns corona is the gas consisting of hot ionized gas or plasma surrounding the Sun, extending millions of km into space. It is normally made invisible by the Suns glare, but is clearly visible during a solar eclipse. The temperature of the corona is...