Understanding Quantum Mechanics: From Classical Mechanics to Schrödinger Equation - Prof. , Study notes of Physical Chemistry

Download Understanding Quantum Mechanics: From Classical Mechanics to Schrödinger Equation - Prof. and more Study notes Physical Chemistry in PDF only on Docsity! 1 Classical Mechanics Newton’s Second Law tells us how to find information about the future state of a Classical Mechanical system from its present state: 2 2 dt xd mmaF  2 2 dt xd dt dx dt d dt dv a       Acceleration: 2 Development of Quantum Mechanics Starting Point: Classical Wave Mechanics Wave-like nature of matter suggests that a wave equation should be used to describe atoms, molecules, and electrons. wave vector  2 k angular frequency  2 phase shift In general, for a wave propagating in a non-dispersive medium (at velocity v): Classical Nondispersive Wave Equation 5 Quantum Mechanics Strategy to understanding QM: Postulate the basic principles and use these postulates to deduce experimentally testable consequences. e.g. Energy levels of atoms To describe the state of a system in Quantum Mechanics, postulate the existence of a function of the particles coordinates called the wavefunction (Y).  tx,YY For a one dimensional system: Y contains all of the possible information about the system. 6 The Schrödinger Equation        txtxV x tx mt tx i ,, , 2 , 2 22 Y  Y   Y   Time-dependent Schrödinger Equation:        xExxV dx xd m    2 22 2  Time-independent Schrödinger Equation: TISE 2 h Note: for a particle of mass m moving in one dimension x with energy E In 1926, Austrian Physicist Erwin Schrödinger used the Classical Nondispersive Wave Equation along with the de Broglie Relation to formulate a Quantum Mechanical wave equation. 7 The Postulates of Quantum Mechanics Postulate 2: Every Observable has a Corresponding Operator Quantum mechanics can be expressed in terms of six postulates which summarize the principle tenets of quantum mechanics. Postulate 1: The Physical Meaning of the Wavefunction Postulate 3: The Result of any Individual Measurement Postulate 4: The Expectation Value Postulate 5: The Time Evolution of a Quantum Mechanical System Postulate 6: The Symmetry of the Wavefunction WRT Parity 10 Postulate 1: The Physical Meaning of the Wavefunction The Properties of the Wavefunction (Y or ) i.  must be normalized. In order for the wavefunction  to be a plausible solution to the Schrödinger of real system (i.e. physical), the following restrictions hold: ii.  must be finite. iii.  must be continuous. iv.  must be single-valued. 11 Postulate 1: The Physical Meaning of the Wavefunction The Properties of the Wavefunction (Y or ) (i) Normalization        xExxV dx xd m    2 22 2  TISE If  is a solution to this equation, than N is also a solution, where N is any normalization constant. For the normalized wavefunction N, the probability of finding the particle in the region of space dx is given by (N)* (N)dx. The sum over all space of these individual probabilities must equal 1: 1*2    dxN  What is the value of the normalization constant, N? (*complex conjugate) 12 Postulate 1: The Physical Meaning of the Wavefunction The Properties of the Wavefunction (Y or ) i.  must be normalized. ii.  must be finite. iii.  must be continuous. iv.  must be single-valued. In order for the wavefunction  to be a plausible solution to the Schrödinger of real system (i.e. physical), the following restrictions hold: 15 Postulate 2: Every Observable has a Corresponding Operator In quantum mechanics, the eigenfunctions corresponding to different eigenavlues of the same operator are orthogonal. All quantum mechanical operators belong to a mathematical class of called Hermitian operators that have real eigenvalues. 0*     dji unless i = j    dOdO jiji       *ˆˆ* 0*     dji If i and j are wavefunctions for the same system but correspond to two different energies, we know they must be orthogonal, so: 16 Postulate 2: Every Observable has a Corresponding Operator Some quantum mechanical operators do not generate eigenfunction/eigenavlue equations. kxAcos2 Example: What is the momentum of a particle moving in 1D? Assume the following form of the wavefunction: )( ikxikx eeA   To find the linear momentum we must operate on the wavefunction with :xp̂ kxA dx d i px cos2ˆ   kxA i kx dx d A i sin2 cos2     In general, when  is not an eigenfunction of an operator, the property to which the operator corresponds does not have a definite value. Recall )( 2 1 cos ikxikxeex   17 Postulate 3: The Result of any Individual Measurement In any single measurement of the observable that corresponds to the operator , Ô the only values that will ever be measured are the eigenvalues of that operator.    txEtxH nnn ,,ˆ YY Solving the Schrödinger equation means finding the complete set of eigenfunctions (Yn) and eigenvalues (En) of the Hamiltonian operator. Some properties of a complete set:  k kknn cccc  2211 1. A set of functions is complete when any arbitrary function can be expressed as a superposition (linear combination) of them: 2. In a single measurement, only one of the eigenvalues corresponding to the particular k that contributes to the superposition will be found.