Modern Physics lecture 3

Louis de Broglie

Wave Properties of Matter In 1923 Louis de Broglie postulated that perhaps matter exhibits the same “duality” that light exhibits In 1923 Louis de Broglie postulated that perhaps matter exhibits the same “duality” that light exhibits Perhaps all matter has both characteristics as well Perhaps all matter has both characteristics as well Previously we saw that, for photons, Previously we saw that, for photons, Which says that the wavelength of light is related to its momentum Which says that the wavelength of light is related to its momentum Making the same comparison for matter we find… Making the same comparison for matter we find…

Quantum mechanics Wave-particle duality Wave-particle duality  Waves and particles have interchangeable properties  This is an example of a system with complementary properties The mechanics for dealing with systems when these properties become important is called “Quantum Mechanics” The mechanics for dealing with systems when these properties become important is called “Quantum Mechanics”

The Uncertainty Principle Measurement disturbes the system

The Uncertainty Principle Classical physics Classical physics  Measurement uncertainty is due to limitations of the measurement apparatus  There is no limit in principle to how accurate a measurement can be made Quantum Mechanics Quantum Mechanics  There is a fundamental limit to the accuracy of a measurement determined by the Heisenberg uncertainty principle  If a measurement of position is made with precision  x and a simultaneous measurement of linear momentum is made with precision  p, then the product of the two uncertainties can never be less than h/4 

The Uncertainty Principle In other words: In other words:  It is physically impossible to measure simultaneously the exact position and linear momentum of a particle These properties are called “complementary” These properties are called “complementary”  That is only the value of one property can be known at a time  Some examples of complementary properties are  Which way / Interference in a double slit experiment  Position / Momentum (  x  p > h/4  )  Energy / Time (  E  t > h/4  )  Amplitude / Phase

Schrödinger Wave Equation The Schrödinger wave equation is one of the most powerful techniques for solving problems in quantum physics The Schrödinger wave equation is one of the most powerful techniques for solving problems in quantum physics In general the equation is applied in three dimensions of space as well as time In general the equation is applied in three dimensions of space as well as time For simplicity we will consider only the one dimensional, time independent case For simplicity we will consider only the one dimensional, time independent case The wave equation for a wave of displacement y and velocity v is given by The wave equation for a wave of displacement y and velocity v is given by

Erwin Schrödinger

Solution to the Wave equation We consider a trial solution by substituting We consider a trial solution by substituting y (x, t ) =   (x ) sin(  t ) into the wave equation By making this substitution we find that Where  / v = 2  / and p = h/ Thus    /  v 2  (2  / ) 

Energy and the Schrödinger Equation Consider the total energy Consider the total energy Total energy E = Kinetic energy + Potential Energy E = m v 2 /2 +U E = m v 2 /2 +U E = p  /(2m ) +U E = p  /(2m ) +U Reorganise equation to give Reorganise equation to give p 2 = 2 m (E - U ) From equation on previous slide we get From equation on previous slide we get Going back to the wave equation we have This is the time-independent Schrödinger wave equation in one dimension

Wave equations for probabilities In 1926 Erwin Schroedinger proposed a wave equation that describes how matter waves (or the wave function) propagate in space and time In 1926 Erwin Schroedinger proposed a wave equation that describes how matter waves (or the wave function) propagate in space and time The wave function contains all of the information that can be known about a particle The wave function contains all of the information that can be known about a particle

Solution to the SWE The solutions  (x) are called the STATIONARY STATES of the system The solutions  (x) are called the STATIONARY STATES of the system The equation is solved by imposing BOUNDARY CONDITIONS The equation is solved by imposing BOUNDARY CONDITIONS The imposition of these conditions leads naturally to energy levels The imposition of these conditions leads naturally to energy levels If we set If we set We get the same results as Bohr for the energy levels of the one electron atom The SWE gives a very general way of solving problems in quantum physics

Wave Function In quantum mechanics, matter waves are described by a complex valued wave function,  In quantum mechanics, matter waves are described by a complex valued wave function,  The absolute square gives the probability of finding the particle at some point in space The absolute square gives the probability of finding the particle at some point in space This leads to an interpretation of the double slit experiment This leads to an interpretation of the double slit experiment

Interpretation of the Wavefunction Max Born suggested that  was the PROBABILITY AMPLITUDE of finding the particle per unit volume Max Born suggested that  was the PROBABILITY AMPLITUDE of finding the particle per unit volume Thus Thus |   | 2  dV =        dV |   | 2  dV =        dV (     designates complex conjugate ) is the probability of finding the particle within the volume dV (     designates complex conjugate ) is the probability of finding the particle within the volume dV The quantity |   | 2  is called the PROBABILITY DENSITY The quantity |   | 2  is called the PROBABILITY DENSITY Since the chance of finding the particle somewhere in space is unity we have Since the chance of finding the particle somewhere in space is unity we have When this condition is satisfied we say that the wavefunction is NORMALISED

Max Born

Probability and Quantum Physics In quantum physics (or quantum mechanics) we deal with probabilities of particles being at some point in space at some time In quantum physics (or quantum mechanics) we deal with probabilities of particles being at some point in space at some time We cannot specify the precise location of the particle in space and time We cannot specify the precise location of the particle in space and time We deal with averages of physical properties We deal with averages of physical properties Particles passing through a slit will form a diffraction pattern Particles passing through a slit will form a diffraction pattern Any given particle can fall at any point on the receiving screen Any given particle can fall at any point on the receiving screen It is only by building up a picture based on many observations that we can produce a clear diffraction pattern It is only by building up a picture based on many observations that we can produce a clear diffraction pattern

Wave Mechanics We can solve very simple problems in quantum physics using the SWE We can solve very simple problems in quantum physics using the SWE This is sometimes called WAVE MECHANICS This is sometimes called WAVE MECHANICS There are very few problems that can be solved exactly There are very few problems that can be solved exactly Approximation methods have to be used Approximation methods have to be used The simplest problem that we can solve is that of a particle in a box The simplest problem that we can solve is that of a particle in a box This is sometimes called a particle in an infinite potential well This is sometimes called a particle in an infinite potential well This problem has recently become significant as it can be applied to laser diodes like the ones used in CD players This problem has recently become significant as it can be applied to laser diodes like the ones used in CD players

Wave functions The wave function of a free particle moving along the x-axis is given by The wave function of a free particle moving along the x-axis is given by This represents a snap-shot of the wave function at a particular time This represents a snap-shot of the wave function at a particular time We cannot, however, measure , we can only measure |  | 2, the probability density We cannot, however, measure , we can only measure |  | 2, the probability density