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Schroedinger equation Basic postulates of quantum mechanics. Operators: Hermitian operators, commutators State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness eigenvalues and expectation values of operators Time independent Schroedinger equation and stationary states. Probability current. basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Schroedinger equation. Schroedinger equation is a wave equation, which links time evolution of the wave function of the state to the Hamiltonian of the state. For most of systems Hamiltonian “represents” total energy of the system T+V= kinetic +potential. Hamiltonian is defined also classically, and equations of motions for classical systems can be written using derivatives of the Hamiltonian. Classically there is no need for a concept of the wave function of the state, as any state can be totally specified by giving momentum and positions of all particles. basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ - First I remind you about a flat wave . This is the wave function describing a free particle. - I will show that the flat wave is a solution of the free Schroedinger equation. - It useful to “test” operators and properties of wave functions on a flat wave to understand what they really mean. basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ A non-relativistic particle has the following Hamiltonian-> energy: 2 p H =[ V r ] 2m where kinetic energy is m v 2 p 2 Ek= = 2 2m we know that a free particle ( propagating in a place without potential) can be described by a flat wave or a combination of flat waves- wave packet m r , t = A e i p r −Et h r , t =2 h −3 2 h h =E ,=h / p , ℏ= 2 where ∫ p e i p r −Et h 3 d p we note that: −3 i p r −Et ∂ r , t −i 3 =2 h 2 ∫ p E e h d p ∂t h basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Thus −3 i ∂ r , t p h p r −Et 3 2 −i =2 h ∫ p e d p ∂t h 2m 2 Now remind ourselves Laplace operator ∇ = ∂ 2 ∂ 2 ∂ 2 ∂x ∂y ∂z 2 2 2 ∂ r , t 2 ∂x 2 −3 2 =−2 ℏ and note that for example : 1 p p x e 2 ∫ 2 i p r −Et ℏ 3 d p ℏ i p r −Et 1 2 2 ℏ 2 ∇ r , t =−2 ℏ 2 ∫ p p e d3 p ℏ thus 2 −3 ∂ h 2 ih =[− ∇ ] ∂t 2m 2 basic principles 20/01 This is Schroedinger equation for free particle Anna Lipniacka www.ift.uib.no/~lipniack/ We can also note that differentiating over x (for example): −3 i p r −Et h ∂ r , t 3 2 h =−2 h ∫ p p x e d p i ∂x and define momentum operators h ∂ h ∂ h ∂ p x ≝ , p y ≝ , p z ≝ i ∂x i ∂y i ∂z 2 2 ∂ p ∂ h 2 ih = H =[ ] ih =[− ∇ ] ∂t 2m ∂t 2m Natural extension to the situation with potential ( non-free particle) 2 ∂ p ih = H =[ V r ] ∂t 2m ∂ h2 2 ih =[− ∇ V r ] ∂t 2m basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Postulates of Quantum Mechanics A ) For every “classical” observable a linear operator. It can depend on momentum and position operators. Momentum operator (x) is proportional to derivative over x F q q .. q , p p .. p : q =q , p = ℏ ∂ 1, 2, n 1, 2, n n n n i ∂ qn B) A system is fully described by a wave-function which fulfills wave equation, Schroedinger 2 for non-relativistic Hamiltonian C) ∂ p ih = H =[ V r ] ∂t 2m expectation value of an operator F < F >=∫ F d * corresponds to the expectation ( mean value) of the measurement result of the variable F taken over a big number of independent measurements (this is more difficult then it sounds) D) the only possible results of single measurements of the variable F are the eigenvalues of the operator F F = f f = constant (example of spin ½ , z projections ) (after a measurement the system “collapses” to the state with well defined variable f) basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ What's new here ? You have heard this before on “kvantefysikk og statistik mekanikk” The difference is that now we are trying to write our wave function in a more general way. It can be a function of space variables, or momentum or perhaps even more general variables ( and time). In practice we will start with space representation, then we will discuss momentum representation, and then other- general Dirac representation of the sate and vector representation ( matrix mechanics) basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Ad C. < F >=∫ F d * d =dq 1 .... d q n In general we integrate over the set variables the wave function of the system is defined on, and normalized on. In “space representation” of quantum mechanics we discuss right now this is space coordinates. For example an expectation value of a position of a particle will be: < r t >=∫∫∫ r , t r r , t d r * 3 < r t >=∫∫∫ r ∣ r , t ∣ d r 2 3 what about momentum ? 1=∫∫∫ ∣ r , t ∣ d r 2 “proper normalization” 3 This is probability density to find the state in location r basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Hermitian operators. The expectation values of an operator representing any real variable must be real * * < F >=∫ F d = ∫ F d =< F > * Definition of hermitian operator: The operator F is hermitian if : * ∫ F 2 d =∫ 2 F 1 d * 1 If the operator is hermitian its expectation value is REAL- that is a requirement for operators associated with observables Definition of adjoined operator to F * ∫ F 2 d =∫ 2 F 1 d * 1 Hermitian operators are self-adjoined meaning basic principles 20/01 : F =F example, check p_x operator. Anna Lipniacka www.ift.uib.no/~lipniack/ Ad hermitian, prove of hermeticity condition * < F >=∫ F d =∫ F d * (alpha, any constant number- take real functions PSI1, PSI2 ia * ia ia ia * e F e d = e F e F d ∫ 1 ∫ 1 2 1 2 2 1 2 * ia * −ia * * F d e F e F F 2 d = ∫ 1 ∫ 1 ∫ 2 1 2 1 ∫ 2 * d = F * d e−ia F * eia F * F ∫ 1 ∫ 1 1 2 ∫ 2 1 ∫ 2 2 * ∫ F 2 d =∫ 2 F 1 d * 1 * ∫ F 1 d =∫ 1 F 2 d * 2 basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Commutator of operators Commutators: are important! for example variables who's operators commute can be measured simultaneously for the system. ,B ]= A B− B A [A Lets check [ x , p x ]= x p x − p x x ℏ ∂ ℏ ∂ x x p x − p x x =x − i ∂x i ∂x ℏ ∂ ℏ ∂ ℏ =x − x − =ℏ i i ∂ x i ∂ x i x position variable, and p_x do not commute. [ x , p x ]=i ℏ What does that mean for subsequent measurements of px and x ? basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Eigenfunctions and eigenvalues: The result of an operator working on a function is usually a different function . If there exist a set of functions that : F n = f n n we call them eigenfunctions of the operator F, and fn are eigenvalues (constants) The set of eigenvalues ( or spectrum of the operator) can be discrete , continuos or mixed. Example of continuos- energy operator (Hamiltonian) for a free particle, discrete= hamiltonia for harmonic oscillator, mixed- hamiltonian for hydrogen atom. ℏ ∂ p x = i ∂x ℏ ∂ ifx f = f f f =const ∗exp i ∂x ℏ That's the form of eigenfunctions of momentum operator. They are not quadraticaly integrable and have to be normalized in a different way. basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Expectation value vs. eigenvalue of an operator. < F >=∫ F d * The expectation ( mean value) of the measurement of the variable F taken over a big number of independent measurements ( in practice over large number of identically prepared states- an ensamble ). But the only possible results of measurements of the variable F are the eigenvalues of the operator F (example of spin ½ , z projections ) F = f If the system is in the state described by the eigenfunction of the operator ,the expectation value of the measurement is equal to the eigenvalue. As the eigenvalue is the only possible measurement results, that is what we will always get ! * * < F >=∫ F d =∫ n F n d = f n∫ n n = f n * for the eigenstate number “n” This proof is straightforward for normalized quadratic-integrable eigenfunctions, can be also proved for different type of normalization basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Orthogonality of eigenfunctions For a hermetic operator, eigenfunctions corresponding to different eigenvalues are orthogonal * * ∫ n F m d =∫ m F n d * * * f m∫ n m d = f n∫ m n d * f m − f n ∫ n m d =0 if f m ≠ f n then Dirac notation: ∫ * n m d =〈 n∣m〉 ∫ m d =0 Scalar product ∫ 〈 n∣F m〉=〈 F n∣m〉 basic principles 20/01 * n * n F m d =〈 n∣F m〉 Hermitian operator in Dirac notation Anna Lipniacka www.ift.uib.no/~lipniack/ We try to normalize our set of eigenfunctions of an operator – typically choosing a constant to multiply the functions with. ∫ * n m d = nm 0 for n = m , orthogonality 1 for n=m , normalization Normalization of eigenfunctions with continuous spectrum of eigenvalues has to be a bit different- functions are not “quadratically integrable” ∫ f ' f d = f ' − f =< f' | f> * ifx f x =c∗exp ℏ Example ∞ 1 ∫ ' x f x d x= 2 ℏ −∞ ∫e * f C must be basic principles 20/01 c= 1 2 ℏ ix Dirac delta momentum eigenfunctions, here just 1-dim f − f ' ℏ dx= f ' − f to get the normalization right Anna Lipniacka www.ift.uib.no/~lipniack/ More about Dirac delta function: Dirac delta function : ∞ ∞ −∞ −∞ ∫ f x x −x ' dx = f x ' ∫ f x x dx= f 0 ∞ ∫ 1∗ x dx =1 properties : −∞ possible representation: ∞ 1 x =lim 0 x =lim exp ixy − y dy=lim ∣ ∣ ∫ 0 2 2 2 −∞ x ∞ 1 x = ∫ exp ixy dy 2 −∞ basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ x = x 2 2 =0.01 =0.1 =0.5 x basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Completeness We assume that every “reasonable” , quadratically integrable function can be expressed as a linear combination of eigenfunctions of a hermitian operator F g=∑ c i i | g >=∑ c i |i >=∑ <i | g> |i > i i i for orthonormal set of eigenfunctions the coefficients are a scalar product of the function in question and appropriate eigenfunction ∫ n g d =∑ c i∫ n i d =c n c n=∫ n g d =<n | g> * * * i For eigenfunctions with continuous spectrum of eigenvalues we can represent a function g in the following way : (eigenfunctions have to form a complete basis set) g=∫ c f f df | g >=∫ df c f | f >=∫ df <f| g> | f > ∫ f ' f d = f ' − f =< f' | f> * then we have : c f =∫ g d =< f | g> * f we must also have : basic principles 20/01 ∫ * f r ' f r d f = r ' −r Anna Lipniacka www.ift.uib.no/~lipniack/ Completeness, continuation: Lets now consider that our functions are normalized in the normal “space”, so ∫ f ' f d =∫ f ' r f r d r = f ' x − f x f ' y − f y f ' z− f z * * 3 etc.. c n =∫ n r ' g r ' d r ' g r =∑ c i i r * i g r =∫ g r ' ∑ i r ' i r d r ' * i 3 3 ∑ i r ' i r = r −r ' * i 3-dim Dirac delta we must in analogy have for continuous spectrum: basic principles 20/01 ∫ f r ' f r d f = r ' −r * Anna Lipniacka www.ift.uib.no/~lipniack/ What is the interpretation of expansion coefficients cn ? We see that modulus squared of an expansion coefficient will correspond to a probability to measure certain eigenvalue: Lets take spectrum of eigenfunctions of given operator F and expand a function of state (g) into it : g=∑ c i i and check what is the expectation value of the operator F for the state described by g i * * < F >=∫ g F g d =∫ ∑ c n n F ∑ c i i * n i < F >=∫ ∑ c n n ∑ f i c i i =∑ ∑ c n * c i f i∫ n i d * * n * i n i < F >=∑ ∑ c n * c i f i i,n =∑ ∣c i∣ f i 2 n i i but we know that from interpretation of expectation value we must have < F >=∑ P i f i i basic principles 20/01 where P_i is the probability that the value f_i will be measured. Anna Lipniacka www.ift.uib.no/~lipniack/ Thus we have. The probability that measuring observable associated with F on a state described by a wave function “g” will give a result f_i is the following: P i =∣c i∣ =∣∫ g d ∣ 2 F i = f i i 2 * i where For continuos spectrum of F we can prove in analogy that: 〈 F 〉=∫ f ∣c f ∣ df 2 The probability that measuring observable associated with F on a state described by a wave function “g” will give a result beetween f and f+df is P f df =∣c f ∣ df =∣∫ g d ∣ df 2 basic principles 20/01 * f 2 where F f = f f Anna Lipniacka www.ift.uib.no/~lipniack/ Stationary states: If the Hamiltonian does not contain time explicitly we can try to separate the solutions in to time dependent part and coordinates dependent part. We obtain that there is new constant involved proportional to the time derivative of the time dependent function divided by the function itself. We call it ENERGY. We obtain time-independent Shroedinger equation for the part which does not depend on time. t , r ≝ r T t dT t / dt iℏ =const ≝E T t r =E r H −i Et / ℏ T =C e ∂ T t h2 2 ih r =[− ∇ r V r r ]T t ∂t 2m What “stationary” means in practice ? Expectation values of operators do not depend on time. ( for normal operators which do not contain time derivatives ) Prove it ! basic principles 20/01 Anna Lipniacka www.ift.uib.no/~lipniack/ Probability current : Probability interpretation for the particle wave function: modulus square of it is a probability to find a particle in a given place (probability density). THUS: Integral over space has to give 1. However locally spacial probability density can change with time. We can define the probability current, useful when discussing movement of particles ∂ r , t ∂ ∂ ℏ ℏ * ∂ * 2 2 * * * = = =i ∇ − ∇ =−i ∇ ∇ − ∇ ∂t ∂t ∂t ∂t 2m 2m * * change of probability density with time (at a given place) is related to the out-flow of the current. ∂ ℏ =−i ∇ ∇ * − * ∇ =−∇ j ∂t 2m j =i ℏ ∇ * − * ∇ =ℜ * i ℏ ∇ 2m m ∂ h 2 ih =[− ∇ V r ] ∂t 2m 2 basic principles 20/01 ∂ h 2 * ih =[− ∇ V r ] ∂t 2m * 2 Anna Lipniacka www.ift.uib.no/~lipniack/