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Lecture 8 The Pauli exclusion principle Particle in a box, quantum pressure in solids Inner products, Dirac notation The Heisenberg uncertainty principle Objectives Learn about the Pauli exclusion principle and how it leads to the creation of energy bands. Learn about quantum pressure, a result of the Pauli exclusion principle, and how it prevents the collapse of solids. Learn how to construct a Heisenberg uncertainty relation for two observables. Bosons and fermions Suppose we have two identical particles, e.g. electrons. The wavefunction for a two particle system is a function of the coordinates of particle 1, x 1 , the coordinates of particle 2, x 2 and the time, i.e. x 1, x 2, t . Such a wavefunction is governed by the Schrödinger equation as before and the energy is the total energy of the system. If particle 1 is in a state a x 1 and particle 2 in a state b x 2 then (ignoring time) x 1, x 2 =a x 1 b x 2 , i.e. x 1, x 2 is the product of the two wavefunctions. However, electrons are indistinguishable from one-another in the sense that we cannot create an experiment to tell two electrons apart, we can put x 1, x 2 = x 2, x 1 . We must create a wavefunction that isnt specific as to which electron is at which location. We write ± x 1, x 2 = A[a x 1 b x 2 ±b x 1 a x 2 ] . Bosons and fermions II We classify particles into two types: bosons and fermions. Bosons: particles such as photons and mesons, these have integer spin and use the plus sign. Fermions: particles such as electrons and protons, these have half-integer spin and take the minus sign. Suppose we have two electrons with wavefunctions a and b , then - x 1, x 2 = A[a x 1 b x 2 b x 1 a x 2 ] . If a =b then - x 1, x 2 =0 , i.e. such a state cannot exist. This is known as the Pauli exclusion principle. The Pauli exclusion principle applies to particles whose wavefunctions overlap. Note that we have not considered the effect of spin. Electrons can have two possible values of spin (+½ or -½), so the Pauli exclusion principle in fact demands unique values of x s , where s is a spinor describing the spin of a particle. The Stern-Gerlach experiment Silver atoms are emitted from a hot oven and split into two beams. Silver has a single outer electron with zero orbital angular momentum. The spin of the outer electron is responsible for the splitting. The Stern-Gerlach experiment II The silver atoms have zero angular momentum, so there should be no interaction with the magnetic field (no moving charge = no magnetic moment). This means that the pattern on the photographic plate should be diffuse as each atom would have a random dipole orientation. In fact, the pattern on the plate forms two distinct parts. This suggests that the atoms have just two possible orientations of the magnetic moment. The magnetic moment arises from the spin of the outer electron. We can imagine two kinds of angular momentum: orbital and spin. For example the earth has orbital angular momentum due to its motion around the sun. It also has spin angular momentum due to its rotation about the north-south axis. In quantum mechanics, spin is an intrinsic property of a particle, like mass or charge. Band structure As atoms become closer together the wavefunctions of the electrons begin to overlap. The Pauli exclusion principle results in a smearing of the allowed electron energies into bands. For some elements band gaps are formed, electrons cannot have energies in the band gap region. Energy bands vs. atomic separation, r, in semiconductors Free electron gas In the solid state, in particular in metals, the most loosely bound electrons are free to move around. The electron gas theory of Sommerfeld considers these electrons as free particles in a box. We start by assuming the electrons are confined to a box of size lx, ly and lz. They are free to move inside the box but cannot escape it. This is similar to the infinite square well model, except that we have three dimensions instead of one. The potential acting on the electron is V x , y , z=0 for 0 xl x , 0 yl y and 0 zl z and V x , y , z= otherwise. 2 2 The Schrödinger equation is = E , we can separate the 2m wavefunction into x, y and z components x , y , z= X x Y y Z z . 2 2 2 d2 X d2Y d2 Z =Ex X , =E y Y =E z Z So and with 2 m dx 2 2 m dy 2 2 m dz 2 E = E x E y E z . Solutions to the Schrödinger equation Let k x 2 m Ex = , ky 2m Ey = and k z 2 m Ez = 2 d X 2 =k so that x X etc. 2 dx X x = Ax sin k x x B x cos k x x , The general solutions are Y y = A y sin k y y B y cos k y y and Z z= Az sin k z z B z cos k z z . X 0=Y 0=Z 0=0 and We know that at the boundaries X l x =Y l y =Z l z =0 so B x = B y = B z=0 . Therefore k x l x =n x , k y l y =n y and k z l z =n z , where nx, ny and nz are integers. l 2 2 nx x dx=1 To find the normalisation coefficients we must solve Ax sin lx 0 2 2 lx =1 and Ax = So Ax etc. The wavefunction for the electron becomes lx 2 x n n x y nx ny nz 8 sin x sin y sin z . n x , y , z= l xl y lz lx ly lz z k-space The energy of the wavefunction is given by 2 2 2 2 2 n x n y n z 2 k 2 where k is the magnitude of the wave En ,n ,n = 2 2 = 2 2m lx ly lz 2m vector k = k x , k y , k z . Each k vector can be associated with a volume of kspace x y z The volume associated with each k vector 3 3 = . is lx l yl z V Because of the Pauli exclusion principle only two electrons (spin-up and spindown) can occupy any given state. Quantum pressure Suppose an object contains N atoms, each of which contributes q free electrons. The k-space will be filled, starting with the lowest energy states, and will have the shape of an octant of a sphere (because kx, ky and kz are positive). 3 Nq 1 4 The volume of the k-space is given by (the number of k 3f = 8 3 2 V states times the volume of each state), kf is the radius of the sphere. Nq 2 1/3 , then k f =3 . V The surface of the sphere of k-space is called the Fermi surface. The 2 2 k f 2 energy of the Fermi surface is E f = = 3 2 2 / 3 . 2m 2 m We define the free electron density as = Quantum pressure II We can calculate the total energy of the electron gas by considering the energy of a spherical shell of thickness dk. The volume of the shell is 2 k dk 1 2 , and the number of states in the shell is V k = 4 k dk = 8 2 2 2 2V k Vk 2 dk k nk = 3 = . Each state has an energy , so the energy of the /V 2 2m 2 2 2 k Vk dk dE= . The total energy is then shell is 2 m 2 k 2 5 2 2 5/3 2 V k 3 Nq V f 2/ 3 4 = V k dk = E= . 2 2 2 2m 0 10 m 10 m f If the box expands by an amount dV then the energy changes by an 2 2 5 /3 2 3 Nq 2 dV 5/ 3 dE= V dV = E amount , i.e. the energy decreases. 2 3 3 V 10 m Conversely, if the box shrinks the energy increases. Quantum pressure III We can conceive of a quantum pressure which acts on the Fermi surface and prevents the box from collapsing. If we try to shrink the size of the box the work done is dW = PdV , so 2 2 5/3 3 Nq 2 E 2 5/ 3 P= = V , the quantum pressure resists the shrinking 3 V 3 10 m 2 of the box. The quantum pressure is sometimes called the degeneracy pressure and is a result of the requirement that the electrons must have different wavefunctions. The quantum pressure is one reason why a cold solid object cannot collapse. Quantum pressure prevents the collapse of cold objects Images of ultracold Li atom clouds. Li-7 is a boson and Li-6 is a fermion. The Pauli exclusion principle leads to a pressure which prevents the fermion cloud from shrinking. The same effect prevents white dwarf stars from collapsing under their own gravity. Inner products (scalar product) We saw before that vectors can be written as | . We can also consider functions in a similar way. Dirac introduced a bracket notation to represent the scalar product of two vectors or functions. * * * In Dirac notation we can write ==a1 b1 a 2 b 2 ... an bn . Suppose | =1 i x and | =1i y , then =0 , i.e. the two vectors are orthogonal. Also =1 i1i=2, = , which is the norm (length) of the vector. For a set of mutually orthogonal, normalised vectors we can write i j = ij , where ij =1 for i= j and ij =0 for i j . For two functions f x and g x the inner product can be written as f g = f x * g x dx . For this to work properly we must set the condition that f x2 dx . Inner products II In Dirac notation the inner product of two functions is written as 1 2 . The symbol | 2 is known as a ket whilst the symbol 1 | is known as a bra. From the definition we can write 1 2 = 21 * If c is a complex number we have 1c 2 =c 1 2 * c 1 2 =c 1 2 3 1 2 = 3 1 3 2 If the two functions are orthogonal then 1 2 =0 If the wavefunctions are normalised then 1 1 = 2 2 =1 There are some transformations T such that T = T , these are known as Hermitian transformations. Heisenberg uncertainty principle We can write the expectation values of x and p in Dirac notation as x =x and p = p . The deviation of a measurement of x or p from x or p is x = x x and p = p p . The corresponding operators are x x and p p . 2 2 So x = x x and p = p p . 2 2 Both of these operators are Hermitian so x = x x x x 2 p = p p p p . 2 2 Let x= x x and p= p p . Then x = xx and p = p p . 2 2 and 2 2 The Schwartz inequality says x p = xx p p x p . For 2 a complex 2 2 number 2 z = Re z Im z Im z . z, Im z= 1 z z * 2i and Heisenberg uncertainty principle 2 1 2 2 x p px . So, if z= x p we can write x p 2i It can be shown that x p= x p x p and px = p x x p . So x p px = x p p x =[ x , p ] , where [ x , p ] is the commutator of the two operators. 2 1 2 2 [ x , p ] . So 2 p 2i We know that x = x and p =i , we can use a test function, f(x), to work out [ x , p ] . d d [ x , p ] f x= x i f i xf dx dx df df [ x , p ] f x=i x i x f =i f x and so [ x , p ]=i . dx dx 2 2 1 Therefore and x p . i = 2 2i 2 The same argument holds for other incompatible observables. 2 x 2 p Conclusions The Pauli exclusion principle states that the wavefunctions of fermions in the same state cannot overlap. This leads to the creation of energy bands in solids and a quantum pressure which prevents the collapse of solid materials. The Heisenberg uncertainty principle can be determined for any two incompatible observables from the corresponding operators.