Lecture 4. Reading: Notes and Brennan

Transcription

1 Lecture 4 Postulates of Quantum Mechanics, Operators and Mathematical Basics of Quantum Mechanics Reading: Notes and Brennan.3-.6 ECE Dr. Alan Doolittle

2 Postulates of Quantum Mechanics Classical Mechanics describes the dynamical state variables of a particle as, y, z, p, etc... Quantum Mechanics takes a different approach. QM describes the state of any particle by an abstract Wave Function, (, y,z,t, we will describe in more detail later. We will introduce Five Postulates of Quantum Mechanics and one Governing Equation, known as the Schrödinger Equation. Postulates are hypotheses that can not be proven. If no discrepancies are found in nature, then a postulate becomes an aiom or a true statement that can not be proven. Newton s st and nd Laws and Mawell s equations are aioms you are probably already familiar with. There are also classical static variable such as mass, electronic charge, etc... that do not change during physical processes. ECE Dr. Alan Doolittle

3 Postulates of Quantum Mechanics Postulate The Wave Function, (, y,z,t, fully characterizes a quantum mechanical particle including it s position, movement and temporal prerties. (, y,z,t replaces the dynamical variables used in classical mechanics and fully describes a quantum mechanical particle. The two formalisms for the wave function: When the time dependence is included in the wavefunction, this is called the Schrödinger formulation. We will introduce a term later called an erator which is static in the Schrödinger formulation. In the Heisenberg formulation, the wavefunction is static (invariant in time and the erator has a time dependence. Unless otherwise stated, we will use the Schrödinger formulation ECE Dr. Alan Doolittle

4 Postulates of Quantum Mechanics Postulate The Probability Density Function of a quantum mechanical particle is: (, y,z,t (, y,z,t The probability of finding a particle in the volume between v and v+dv is: (, y,z,t (, y,z,t dv In D, the probability of finding a particle in the interval between & +d is: (, t (, t d Since (, t (, t d is a probability, then the sum of all probabilities must equal one. Thus, 4. (, t (, t d or (, y, z, t (, y, z, t dv This is the Normalization Condition and is very useful in solving problems. If fulfills 4., then is said to be a normalized wavefunction. ECE Dr. Alan Doolittle

5 Postulates of Quantum Mechanics Postulate (cont d (,y,z,t, the particle wave function, is related to the probability of finding a particle at time t in a volume ddydz. Specifically, this probability is: ( [, y, z, t ddydz or ddydz] But since is a probability, or in D (, y, z, t [ d] ddydz Solymar and Walsh figure 3. ECE Dr. Alan Doolittle

6 Postulates of Quantum Mechanics Postulate 3 Every classically obtained dynamical variable can be replaced by an erator that acts on the wave function. An erator is merely the mathematical rule used to describe a certain mathematical eration. For eample, the derivative erator is defined as d/d. The wave function is said to be the erators erand, i.e. what is being acted on. Classical Dynamical Variable Position Potential Energy V( f( Momentum p p Kinetic Energy m QM Operator Representation V( f( h i h m The following table lists the correspondence of the classical dynamical variables and their corresponding quantum mechanical erator: Total Energy (Kinetic + Potential Total Energy (Time Version f(p E E Total Total h m h f i + V( h i t ECE Dr. Alan Doolittle

7 Postulates of Quantum Mechanics Postulate 4 For each dynamical variable, ξ, there eists an epectation value, <ξ> that can be calculated from the wave function and the corresponding erator ξ (see table under postulate 3 for that dynamical variable, ξ. Assuming a normalized, 4. ξ (, t ξ(, t d or ξ y, z, t ξ (, y, z, t dv The epectation value, denoted by the braces <>, is also known as the average value or ensemble average. Brennan goes through a formal derivation on pages 8-9 of the tet and points out that the state of a QM particle is unknown before the measurement. The measurement forces the QM particle to be compartmentalized into a known state. However, multiple measurements can result in multiple state observances, each with a different observable. Given the probability of each state, the epectation value is a weighted average of all possible results weighted by each results probability. (, ECE Dr. Alan Doolittle

8 ECE Dr. Alan Doolittle Postulates of Quantum Mechanics Postulate 4 Eample: Lets assume the wave function of a QM particle is of an observable is given by: We can normalize (see postulate to get the constant A: Then we can calculate the epectation value of, <> as: Note: That the probability of observing the QM particle is 0 at 0 but multiple measurements will average to a net weighted average measurement of 0. otherwise A t 0, ( ( d d or, (, ( 3 A A A d A d t t -

9 Postulates of Quantum Mechanics Postulate 4 As the previous case illustrates, sometimes the average value of an observable is not all we need to know. Particularly, since the integrand of our <> equation was odd, the integrand was 0. (Looking at whether a function is even or odd can often save calculation effort. We may like to also know the standard deviation or the variance (standard deviation squared of the observable around the average value. In statistics, the standard deviation is: σ And since <>0, all we need is: Similarly, we could also calculate other higher order moments of distributions such as skewness, curtosis, etc... σ 3 0 (, t (, t d or d - σ σ ECE Dr. Alan Doolittle

10 Postulates of Quantum Mechanics Postulate 4 Real world, measurable observables need to be real values (i.e. not imaginary / comple. What makes for a real epectation value? To answer this, we need to consider some prerties of erators and devel (work toward the concept of a Hermitian Operator. Eigenfunctions and Eigenvalues: If the effect of an erator acting on an erand (wave function in our case is such that the erand is only modified by a scalar constant, the erand is said to be an Eigenfunction of that erator and the scalar is said to be an Eigenvalue. Eample: Operator ξ (, t λ (, t s Eigenfunction of Operator ξ Operand Eigenvalue ECE Dr. Alan Doolittle

11 Postulates of Quantum Mechanics Postulate 4 Eample: consider the case of an eponential function and the differential erator ξ d d e λ s d d (, t e λ λ s s e λ s Eigenfunction Eigenvalue of derivative Operand erator Eigenfunctions and Eigenvalues are very important in quantum mechanics and will be used etensively. KNOW THEM! RECOGNIZE THEM! LOVE THEM! ECE Dr. Alan Doolittle

12 Linear Operators: ξ Postulates of Quantum Mechanics Postulate 4 An erator is a linear erator if it satisfies the equation c(, t cξ (, t where c is a constant. is a linear erator where as the logarithmic erator log( is not. Unlike the case for classical dynamical values, linear QM erators generally do not commute. Consider: Classical Mechanics : p p Quantum Mechanics : h i d d h i d d (, t p? h (, t i h (, t i? (, t p d d (, t h d + i d (, t (, t ECE Dr. Alan Doolittle

13 Postulates of Quantum Mechanics Postulate 4 If two erators commute, then their physical observables can be known simultaneously. However, if two erators do not commute, there eists an uncertainty relationship between them that defines the relative simultaneous knowledge of their observables. More on this later... ECE Dr. Alan Doolittle

14 Postulates of Quantum Mechanics Postulate 4: A Hermitian Operator (erator has the prerty of Hermiticity results in an epectation value that is real, and thus, meaningful for real world measurements. Another word for a Hermitian Operator is a Self-Adjoint Operator. Let us work our way backwards for the D case: ξ If ξ is real and the dynamical variable is real (i.e. physically meaningful then, ξ ξ This says nothing about ξ Thus taking the conjugate of ξ ξ more generally an erator is said to be hermician if and (, tξ (, tξ (, tξ (, tξ ξ (, td ξ (, td (, td (, td as in general, ξ everything in the above equation, ξ (, tξ (, tξ ξ (, td (, td it satisfies : This equation is simply the case where ECE Dr. Alan Doolittle

15 Postulates of Quantum Mechanics Postulate 4 Consider two important prerties of a Hermitian Operator Eigenvalues of Hermitian Operators are real, and thus, measurable quantities. Proof: Hermiticity states, Consider first the left side, Now consider the right hand side, (, tξ Thus, λ (, tξ (, tξ s λ (, td (, td s λ (, td λ (, tξ s (, tλ s (, t which is only true if (, tλ (, td (, t(, td s s λ s (, td (, td (, td is real. ECE Dr. Alan Doolittle

16 Postulates of Quantum Mechanics Postulate 4 Consider two important prerties of a Hermitian Operator Eigenfunctions corresponding to different and unequal Eigenvalues of a Hermitian Operator are orthogonal. Orthogonal functions of this type are important in QM because we can find a set of functions that spans the entire QM space (known as a basis set without duplicating any information (i.e. having the one function project onto another. Formally, ξ and λ (, t λ This is true if (, t functions (, td ξ λ is hermician and, are (, t 0 orthogonal and ξ if they satisfy : (, t λ (, t See Brennan section.6 for details ECE Dr. Alan Doolittle

17 Probability current density is conserved. Postulates of Quantum Mechanics Postulate 5 If a particle is not being created or destroyed it s integrated probability always remains constant ( for a normalized wave function. However, if the particle is moving, we can define a Probability Current Density and a Probability Continuity equation that describes the particles movement through a Gaussian surface (analogous to electromagnetics. Brennan, section.5 derives this concept in more detail than I wish to discuss and arrives at the following: Define Probability Density ρ Define Probability Current Density (see Brennan section.5 ( r ( r j h mi ( ( r ( r ( r ( r ρ Probability Current Density Continuity Equation ρ j + t 0 ECE Dr. Alan Doolittle

18 Postulates of Quantum Mechanics Postulate 5 Basically this epression states that the wave function of a quantum mechanical particle is a smoothly varying function. In an isotric medium, mathematically, this is stated in a simplified form as: Lim o Lim o (, t and (, t ( Simply stated, the wave function and it s derivative are smoothly varying (no discontinuities. However, in a non-isotric medium (eamples at a heterojunction where the mass of an electron changes on either side of the junction or at an abrupt potential boundary the full continuity equation MUST be used. Be careful as the above isotric simplification is quoted as a Postulate in MANY QM tets but can get you into trouble (see homework problem in non-isotric mediums. When in doubt, use ρ h mi o (, t o, t ρ ( j + 0 ( r ( r j ( r ( r ( r ( r o t ECE Dr. Alan Doolittle

19 Postulates of Quantum Mechanics Eample using these Postulates Consider the eistence of a wave function (Postulate of the form: ( A ( + cos( 0 for for - π -π π π Figure after Fred Schubert with gratitude. ECE Dr. Alan Doolittle

20 Postulates of Quantum Mechanics Eample using these Postulates Consider the eistence of a wave function (Postulate of the form: ( A ( + cos( 0 for for Applying the normalization criteria from Postulate : A A ( + cos ( + cos π ( + cos + cos π A + sin + (, t (, t d A ( 3π A + 4 3π sin - π π -π π d d π π ( 3π ( + cos( 0 for for - π -π π π ECE Dr. Alan Doolittle

21 Postulates of Quantum Mechanics Eample using these Postulates Show that this wave function obeys the Probability Continuity Equation (Postulate 5 at the boundaries +/-π: ( ( + cos( for - π π j 3π 0 for -π Define Probability Density ρ ( + cos + cos ρ t π ( r ( r ρ 3π ρ ρ is independent of time so 0 t so ( ( sin 3π h Probability Current Density j mi h ( cos sin ( cos + + mi 3π 3π 3π Probability Current Density Continuity Equation j + ( ( ( ( ( sin 3 π 0 ECE Dr. Alan Doolittle

22 Postulates of Quantum Mechanics Eample using these Postulates Show that this wave function obeys the smoothness constraint (Postulate 5 for isotric medium at the boundaries +/-π: ( 3π ( + cos( Lim π (or - π 3π 0 0 for 0 for for - π π -π π ( + cos ( + cosπ and -π or π Lim sin π (or - π 3π 0 0 for -π or π 3π sinπ 3π Thus, the wave function and its derivative are both zero and continuous at the boundary (same for -π. ECE Dr. Alan Doolittle

23 Postulates of Quantum Mechanics Eample using these Postulates What is the epected position of the particle? (Postulate 3 ( 3π ( + cos( 0 for for - π -π π π 3π d 3π ( + cos ( + cos 3π π ( + cos + cos π d d Since the function in parenthesis is even and is odd, the product (integrand is odd between symmetric limits of +/-π. Thus, 0 Note: Postulate 4 is not demonstrated because is not an Eigenfunction of any erator. ECE Dr. Alan Doolittle

24 Momentum-Position Space and Transformations Position Space Fourier Transform Momentum Space ( Φ( p πh e ip / h dp Φ ( p ( πh e ip / h d If ( is normalized then so is Φ( p ( ( d Φ ( p Φ( p dp If a state is localized in position,, it is delocalized in momentum, p. This leads us to a fundamental quantum mechanical principle: You can not have infinite precision measurement of position and momentum simultaneously. If the momentum (and thus wavelength from ph/λ is known, the position of the particle is unknown and vice versa. For more detail, see Brennan p. 4-7 ECE Dr. Alan Doolittle

25 Momentum-Position Space and Transformations b ( e Fourier Transform φ( p be b h p The above wave function is a superposition of 7 simple plane waves and creates a net wave function that is localized in space. The red line is the amplitude envele (related to. These types of wave functions are useful in describing particles such as electrons. The wave function envele can be approimated with a Gaussian function in space. Using the Fourier relationship between space and momentum, this can be transformed into a Gaussian in momentum (see Brennan section.3. The widths of the two Gaussians are inversely related. ECE Dr. Alan Doolittle

26 Momentum-Position Space and Transformations b ( e Fourier Transform φ( p be b h p The widths of the two Gaussians are inversely related. Thus, accurate knowledge of position leads to inaccurate knowledge of momentum. ECE Dr. Alan Doolittle

27 Uncertainty And the Heisenberg Uncertainty Principle Due to the probabilistic nature of Quantum Mechanics, uncertainty in measurements is an inherent prerty of quantum mechanical systems. Heisenberg described this in 97. For any two Hermitian erators that do not commute (i.e. B OP A OP A OP B OP there observables A and B can not be simultaneously known (see Brennan section.6 for proof. Thus, there eists an uncertainty relationship between observables whose Hermitian erators do not commute. Note: if erators A and B do commute (i.e. ABBA then the observables associated with erators A and B can be known simultaneously. ECE Dr. Alan Doolittle

28 Uncertainty And the Heisenberg Uncertainty Principle Consider the QM variable, ξ, with uncertainty (standard deviation ξ defined by the variance (square of the standard deviation or the mean deviation, ( ξ <ξ ><ξ> ξ is the standard/mean deviation for the variable ξ from its epectation value <ξ> There are two derivations of the Heisenberg Uncertainty Principle in QM tets. The one in Brennan is more precise, but consider the following derivation for it s insight into the nature of the uncertainty. Brennan s derivation will follow. ECE Dr. Alan Doolittle

29 Uncertainty And the Heisenberg Uncertainty Principle Insightful Derivation: Consider a wave function with a Gaussian distribution defined by: The normalization constant can be determined as, Taking the Fourier transformation of this function into the momentum space we get, ip / h Φ( p ( πh e d Φ Φ ( p πh ( p ( 4π 4 ( A ( ξ A e π ( 4π ( ξ 4 ( 4π 4 ( ξ ( ξ π h ( ξ π h ( ξ ( ξ e ( ξ e e h p ip / h ( ξ d The standard deviation in momentum space is inversely related to the standard deviation in position space. ECE Dr. Alan Doolittle

30 Uncertainty And the Heisenberg Uncertainty Principle Insightful Derivation: The standard deviation in momentum space is inversely related to the standard deviation in position space. Defining: Position Space σ ( ξ and σ p h ( ξ / ( 4π 4 σ X ( σ X ( e Φ( p ( σ X π ( 4π 4 ( σ X e π σ X σ ( and σ ( p p Φ ( p Momentum Space ( p 4π 4 σ p ( σ σ p π ( p 4π 4 ( σ σ p e π e p p σ ( p h σ p ( p ECE Dr. Alan Doolittle

31 Uncertainty And the Heisenberg Uncertainty Principle Precise Derivation: The solution for the Gaussian distribution is eact only for that distribution. However, the assumption of a Gaussian distribution overestimates the uncertainty in general. Brennan (p derives the precise General Heisenberg Uncertainty Relationship using: A If two erators do not commute (i.e. AB BA in general a relationship can be written such that AB-BAiC B Given (A the following prerty proven true by Merzbacher 970: ( A ( B 0.5C Thus, given the position erator,, and momentum erator, p (ħ/i(d/d, Letting this erate on wave function h i p ( p p d d p h i d d ic ic ic ECE Dr. Alan Doolittle

32 Uncertainty And the Heisenberg Uncertainty Principle Precise Derivation (cont d: A If two erators do not commute (i.e. AB BA in general a relationship can be written such that AB-BAiC B Given (A the following prerty proven true by Merzbacher 970: ( A ( B 0.5C p p Letting this erate on wave function ( p p ic h d h d ic i d i d h d h d ( ic i d i d h d h h d ic i d i i d h ic i h C ic ECE Dr. Alan Doolittle

33 Uncertainty And the Heisenberg Uncertainty Principle Precise Derivation (cont d Thus, since ( A ( B 0.5C, with in this case, A and B p, p For this case, the uncertainies associated with is, and p p ( ( p ih h is p must be related to h ( ( p h Note the etra factor of ½. ECE Dr. Alan Doolittle

34 Uncertainty And the Heisenberg Uncertainty Principle Energy Time Derivation: Using: Group Velocity, v group / t ω/ k and the de Broglie relation pħ k and the Planck Relationship Eħ ω: Using Group Velocity Using de Broglie relation Using Planck Relationship ( ( p t ϖ k t ϖ k ( p ( h k ( t( ϖ ( t E h h h h ( t( E h ECE Dr. Alan Doolittle

35 Operators in Momentum Space Postulate 3 (additional information Equivalent momentum space erators for each real space counterpart. Classical Dynamical Variable QM Operator Real Space Representation QM Operator MomentumSpace Representation Position h i p Potential Energy V( V( V h i p f( f( f h i p Momentum p h i p Kinetic Energy p m h m p m f(p h f i f ( p Total Energy (Kinetic + Potential E Total h m + V( p + V h i p m Total Energy (Time Version E Total h i t h i t ECE Dr. Alan Doolittle

36 Dirac Notation There eists a short hand notation known as Dirac notation that simplifies writing of QM equations and is valid in either position or momentum space. ξ or in momentum space, ξ Φ (, tξ (p, tξ (, td Φ(p, tdp Note: The left side, <, is known as a Bra while the right side, >, is known as a ket (derived from the Bracket notation. The ket is by definition, the comple conjugate of the argument. ξ (, tξ (, td Dirac notation is valid in either position or momentum space so the variables,, y, z, t, and p can be tionally left out. ECE Dr. Alan Doolittle

37 Dirac Notation The erator acts on the function on the right side: ξ ξ (, tξ (, td If you need the erator to act on the function to the left, write it eplicitly. ξ (, tξ (, td Note that in this eample, we have commuted the left hand side (erator and with the right hand side (. This is okay. While not all erators commute, any function times another function does commute (i.e. once the erator conjugate has acted on the, the result is merely a function which always commutes. Thus, it is the erator that may or may not commute but the functions that result from an erator acting on a wave function always commute. ECE Dr. Alan Doolittle

38 Dirac Notation The Dirac form of Hermiticity is: (, tξ (, td ξ ξ or ξ ξ (, tξ (, td ECE Dr. Alan Doolittle

39 ECE Dr. Alan Doolittle The Dirac Delta Function is: Dirac Delta Function ( ( ( ( for 0 for o o 0 lim d e o o o o o δ δ δ π σ δ σ σ Some useful prerties of the delta function are: ( ( ( ( ( ( ( ( ( o o o o o o o o f d d d d d f f d f f f a a ( ( ( ( for 0 ( ( o δ δ δ δ δ δ δ δ δ o 0

40 Dirac and Kronecker Delta Function The Dirac Delta Function is: δ ( o for o 0 otherwise The Kronecker Delta Function is: δ ij 0 if i j if i j ECE Dr. Alan Doolittle