Understanding the Time-Indep. Schrödinger Equation & Solutions in Quantum Mechanics, Study notes of Physical Chemistry

Download Understanding the Time-Indep. Schrödinger Equation & Solutions in Quantum Mechanics and more Study notes Physical Chemistry in PDF only on Docsity! CHEM 3510 Fall 2003 Postulates and Principles of Quantum Mechanics The Schrödinger equation – It is the fundamental equation of quantum mechanics. – The solutions of time-independent Schrödinger equation are called stationary state wave functions. Look at time independent Schrödinger equation first. – Schrödinger equation is the equation for finding the wave function of a particle and come up based on idea that if the matter possesses wavelike properties there must be a wave equation that governs them. – Schrödinger equation cannot be demonstrated (can be seen as a fundamental postulate) but it can be understood starting from classical mechanics wave equation: 2 2 22 2 1 t u x u ∂ ∂ = ∂ ∂ v – The solution is: txtxu ωψ cos)(),( = (t dependence: tωcos or or ) )(tT tie υπ2 – Obtained ψ (the spatial amplitude of the matter wave) from the equation: 0)()( 2 2 2 2 =+ x dx xd ψωψ v where λ ππω v22 == v ⇒ 0)(4)( 2 2 2 2 =+ x dx xd ψ λ πψ – Rearrange the equation considering: V m pVKE +=+= 2 2 ⇒ )(2 VEmp −= )(2 VEm h p h − ==λ ⇒ 0)()]([2)( 22 2 =−+ xxVEm dx xd ψψ h ⇒ )()()()( 2 2 22 xExxV dx xd m ψψψ =+− h – This is one-dimensional time-independent Schrödinger equation. – The solutions (wave functions) of this equation are called stationary-state wave functions. CHEM 3510 Fall 2003 – Rewrite the equation and generalize to three dimensions: ),,(),,(),,(),,( 2 2 2 2 2 2 22 zyxEzyxzyxVzyx zyxm ψψψ =+        ∂ ∂ + ∂ ∂ + ∂ ∂ − h – Rewrite including the notation for the Laplacian operator: 22 2 2 2 2 2 ∇=        ∂ ∂ + ∂ ∂ + ∂ ∂ zyx (Laplacian operator) An operator is a symbol that tells you to do something (a mathematical operation) to whatever (function, number, etc) follows the symbol. ⇒ ψψ EzyxV m =         +∇− ),,( 2 2 2h – Rewrite including the notation for the Hamiltonian operator: HzyxV m ˆ),,( 2 2 2 =+∇− h (Hamiltonian operator) ⇒ (Simple form of the Schrödinger equation) ψψ EH =ˆ Eigenvalue problem in Quantum Mechanics – The eigenvalue-eigenfunction problem in quantum mechanics is written in the form: )()(ˆ xxA αφφ = where A is an operator ˆ )(xφ is an eigenfunction or characteristic function α is an eigenvalue or characteristic value – The wave functions are eigenfunctions of the Hamiltonian operator and the total energy is the eigenvalue. Interpretation of the wave function ψ – The * dxxx )()( ψψ is the probability that the particle to be located between x and dxx + (the one-dimensional case). – The function *ψ is the complex conjugate of the wavefunction ψ and is obtained by replacing i to in the wave function i− ψ . In this way the product ψψ * becomes real. CHEM 3510 Fall 2003 Postulate 2 To every observable in classical physics there corresponds a linear, Hermitian operator in quantum mechanics. – An operator is a symbol that tells to do a mathematical operation to whatever follows the symbol. The operators are usually denoted by a capital letter with a little hat over it called a carat (like in Ĥ ). What follows the operator is called operand. – An operator is linear if:  [ ] ( )xfAcxfAcxfcxfcA 22112211 ˆ)(ˆ)()(ˆ +=+ – An operator is Hermitian if it has the property of being linear and if:  dxAdxA ∗ ∫∫ = spaceall 12 spaceall 2 * 1 )ˆ(ˆ ψψψψ (or Af∫ ( ) dxxfAxgdxxgx ∗∗ ∫= spaceallspaceall )](ˆ)[()(ˆ) for any pair of functions 1ψ and 2ψ (or f and g) representing a physical state of a particle. – All the quantum operators can be written starting from the operators in the table below using classical physics formulas: Classical Variable QM Operator Expression for operator x X̂ x xP xP̂ x i xi ∂ ∂ −= ∂ ∂ h h t T̂ (or t ) ˆ t E Ê t i ti ∂ ∂ = ∂ ∂ − h h or ( )zyxV m ,, 2 2 2 +∇− h – Examples: – Kinetic energy: m2 2pK = – One dimensional: 2 222 22 ˆˆ xmm pK ∂ ∂ −== h – Three dimensional: 2 2 2 2 2 2 2 22 22 ˆ ∇−=        ∂ ∂ + ∂ ∂ + ∂ ∂ −= mzyxm K hh CHEM 3510 Fall 2003 – Potential energy: V V=ˆ – Total energy: VKE += – The operator for the total energy is the Hamiltonian Ĥ : V m V zyxm +∇−=+      ∂ ∂ + ∂∂ 2 2 2 2 2 2 2 22 22 ˆˆˆ hVKH   ∂+ ∂ −=+= h – Other properties of operators: Commutation )(ˆˆ)(ˆˆ xfABxfBA ≠ where (( AA = and (B )]ˆ[ˆ)ˆˆ xfBxfB )](ˆ[ˆ)ˆˆ xfABxfA = ⇒ Operators usually do not commute. When for every compatible the operator A and B are said to commute. )(ˆˆ)(ˆˆ xfABxfBA = )(xf – Example: dx ˆ dA = and B 2ˆ x= ⇒ ( ) ( ) ( ) dx xdfxxxfxfBA 22ˆˆ += ⇒ ( ) ( ) dx xdfxxfAB 2ˆˆ = – Rewrite and drop . Define the commutator as: )(xf (commutator of and ABBABA ˆˆˆˆ]ˆ,ˆ[ −=  B̂ ) – The commutator of commuting operators is the zero operator: (where 0 is the zero operator) 0̂ˆˆˆˆ =− ABBA ˆ – A special property of linear operators (for example the operator ) is that a linear combination of two eigenfunctions of the operator with the same eigenvalue is also an eigenfunction of the operator:  – Consider that two eigenfunctions have the same eigenvalue: and (this is a two-fold degeneracy) 11ˆ φφ aA = 22ˆ φφ aA = – Then any linear combination of 1φ and 2φ is also an eigenfunction of :  )(ˆˆ)(ˆ 221122112211 φφφφφφ ccaAcAcccA +=+=+ – Example: )() 22 2 xfm dx x imx− )()( 211 xfcxfc x+ (fd −= has the eigenfunctions and . A linear combination of and , , is also an eigenfunction: imxexf =)(1 exf =)(2 )(12 xf = 1f 2f CHEM 3510 Fall 2003 imximx emcemc dx xfd −−+−= )()( )( 2 2 2 12 12 2 = =+− )( cm 21 2 imximx eec = (fm− )12 2 x Postulate 3 In any measurements of the observable associated with the operator , the only values that will ever be observed are the eigenvalues an, which satisfy the eigenvalue equation:  nnn aA ψψ =ˆ – For an experiment designed to measure the observable associated to , we will find only the values corresponding to the states  naaa ,...,, 21 nψψψ ,...,, 21 and no other values will be observed. – Example: If HA ˆˆ = ⇒ (Schrödinger equation) and only the energies (eigenvalues) will be experimentally observed. nnn EH ψψ =ˆ nE Postulate 4 If a system is in a state described by a normalized wave function ψ , then the average value of the observable corresponding to is given by:  (∫ ∗=〉〈 space all dxÂa ψψ 〉〈a is the symbol for the average value) – Determine the variance (statistical mechanics quantity) of the experiments: – Assume we have: ⇒ )()(ˆ xaxA nnn ψψ = nnn adxxAxa ==〉〈 ∫ ∞ ∞− )(ˆ)(* ψψ – Also: [ ] )()(ˆˆ)(ˆ 22 xaxAx nnn ψψ =AA nψ = ⇒ ( ) 22*2 )( nnnn adxxaxa ==〉〈 ∫ ∞ ∞− ψψ – Variance of the experiments become: 022222 =−=〉〈−〉〈= nna aaaaσ ⇒ The standard deviation aσ is zero so the only values observed are the values. na Postulate 5 The wave function, or state function, of a system evolves in time according to the time-dependent Schrödinger equation: t txitxH ∂ Ψ∂ =Ψ ),(),(ˆ h