Mathematical Formulation of the Superposition Principle

1. Superposition add states together, get new states. Math quantity associated with states must also have this property. Mathematical Formulation of the Superposition Principle Vectors have this property.In real three dimensional space, three basis vectors are sufficient to describe any point in space.Can combine three vectors to get new bases vectors, which are also O.K. under appropriate combination rules. In Q. M., may have many more than three states.Need one vector for each state.May be finite or infinite number of vectors; depends on finite or infinite number of states of the system. Copyright – Michael D. Fayer, 2007

2. Particular ket, A Kets can be multiplied by a complex number and added. complex numbers Can add together any number of kets. If is continuous over some range of x Dirac Call Q.M. vectors ket vectors or just kets. general ket Symbol Copyright – Michael D. Fayer, 2007

3. A ket that is expressible linearly in terms of other kets is dependent on them. is dependent on A set of kets is independent if no one of them is expressible linearly in terms of the others. (In real space, three orthogonal basis vectors are independent.) Dependent and Independent Kets Copyright – Michael D. Fayer, 2007

4. Assume: Each state of a dynamical system at a particular time corresponds to a ket vector , the correspondence being such that if a state results from a superposition of other states, its corresponding ket vector is expressible linearly in terms of the corresponding ket vectors of the other states. Connection between theory and real world Copyright – Michael D. Fayer, 2007

5. If then, can be expressed in terms of and , etc. A state is dependent on other states if its ket is dependent on the corresponding kets of the other states. Order of superposition doesn’t matter. Copyright – Michael D. Fayer, 2007

6. unless then no state. A ket corresponding to a state multiplied by any complex number (other than 0) gives a ket corresponding to the same state. (Unless C = 0) The superposition of a state with itself is the same state. State corresponds to direction of ket vector. The length is irrelevant. The sign is irrelevant. Copyright – Michael D. Fayer, 2007

7.  is a linear function of If To get a number that is a linear function of a vector,take the scalar product with some other vector. Ket number  Copyright – Michael D. Fayer, 2007

8. Other vector Bra symbol particular one Put a bra and a ket together Any complete bracket Incomplete bracket  vector Copyright – Michael D. Fayer, 2007

9. Scalar product with constant times ket. Bra completely defined when its scalar product with every ket is known. Addition of bras defined by scalar product with ket . Multiplication of bra by a constant defined by scalar product with ket . Scalar product with sum of kets. Copyright – Michael D. Fayer, 2007

10. is the complex conjugate of the complex number C. Bra is the complex conjugate of Ket. Assume: There is a one to one correspondence betweenkets and bras such that the bra corresponding to is the sum of the bras corresponding to and that the bra corresponding to is times the bra corresponding to . Theory symmetrical in kets and bras. State of a system can be specified bythe direction of a bra as well as a ket. Copyright – Michael D. Fayer, 2007

11. Since For is a real number becausea number = its complex conjugate is real. Assume Length of a vector is positive unless The complex conjugate of a bracket is the bracketin “reverse order.” Copyright – Michael D. Fayer, 2007

12. Real space vectors – scalar product gives a real number; symmetrical with respect to interchange of order. Kets and Bras – scalar product gives complex number. Interchange of order gives complex conjugate number. Scalar Product Copyright – Michael D. Fayer, 2007

13. Bra and Ket orthogonal if scalar product equals zero. 2 Bras or 2 Kets orthogonal; Orthogonality Two states of a dynamical system are orthogonal if the vectorsrepresenting them are orthogonal. Copyright – Michael D. Fayer, 2007

14. Length doesn’t matter, only direction. May be convenient to have If this is done  States are normalized. Normalization of Bras and Kets Copyright – Michael D. Fayer, 2007

15. g a real number But, can multiply by normalized ket multiply by complex conjugate of still normalized with g real called a “phase factor.” Very important. See later. State of system defined by direction of ket, not length. Normalized ket still not completely specified A ket is a vector – directionNormalized ket – direction and length Copyright – Michael D. Fayer, 2007

16. Kets and bras represent states of a dynamical system, s, p, d, etc states of H atom. Need math relations to work with ket vectors to obtain observables. Ket is a function of ket a is a linear operator. (Underline operator to indicate it is an operator)Linear operators have the properties: complex number Linear Operators A linear operator is completely defined when its application to every ket is known. Copyright – Michael D. Fayer, 2007

17. Additive Multiplication - associative Multiplication isNOTnecessarily commutative in general Rules about linear operators Copyright – Michael D. Fayer, 2007

18. If then really means is a short hand. It does not mean the right and left hand sidesof the equation are equal algebraically. It means have the same result when applied to an arbitrary ket. Only know what an operator does by operating on a ket. Copyright – Michael D. Fayer, 2007

19. Put ket on right of linear operator. Bras and linear operators Put bra on left of linear operator. This is a number. It is a closed bracket number. To see this consider Copyright – Michael D. Fayer, 2007

20. Operate on an arbitrary ket a number Operating on gave , a new ket. Therefore, is an operator. Can also operate on a bra . new bra Kets and Bras as linear operators not closed bracket Will use later; projection operators Copyright – Michael D. Fayer, 2007

21. 1. associative law of multiplication holds2. distributive law holds3. commutative law does not hold4. 5. Have algebra involving bras, kets and linear operators. Assume: The linear operators correspond to the dynamical variables of a physical system. Copyright – Michael D. Fayer, 2007

22. Dynamical Variables coordinatescomponents of velocitymomentumangular momentumenergyetc. Linear Operators  Questions you can ask about a system. For every experimental observable, there is a linear operator. Q. M. dynamical variables not subject to an algebra in whichthe commutative law of multiplication holds.Consequence of Superposition Principle. (Will lead to the Uncertainty Principle.) Copyright – Michael D. Fayer, 2007

23. Already saw that Complex conjugate – reverse order With linear operators complex conjugate of the operator a. complex conjugate of complex conjugate If , the operator is “self-adjoint.” It corresponds to a real dynamical variable. means Conjugate relations Copyright – Michael D. Fayer, 2007

24. The complex conjugate of any product of bras, kets, andlinear operators is the complex conjugate of each factor with factors in reverse order. Copyright – Michael D. Fayer, 2007

25. linear operator ket numbersame ket (real) Know a.want to find and p. eigenvector eigenvalue Apply linear operator to ket, get same ket back multiplied by a number. Can also have is an eigenbra of a. Eigenvalues and Eigenvectors Eigenvalue problem (Mathematical problem in linear algebra.) Copyright – Michael D. Fayer, 2007

26. Observables and Linear Operators Kets (or bras) State of dynamical system. Linear operators Dynamical variables. Question you can ask about the state of a system. Observables Real Dynamical Variables. Observables are the eigenvalues of Hermitian Linear Operators. Hermitian operators Real Dynamical Variables. Hermitian operator If Copyright – Michael D. Fayer, 2007

27. linear operatorobservable valueeigenvector eigenket Observables  Eigenvalues of Hermitian Operators (real eigenvalues) For every observable there is a linear operator (more than one form).The eigenvalues are the numbers you will measure in an experiment. Copyright – Michael D. Fayer, 2007

28. a is multiplied by any number C other than zero, If an eigenvector of eigenvalue. then the product is still an eigenvector with the same (Length doesn't matter, only direction.) A constant commutes with anoperator. A constant can bethought of as a number timesthe identity operator. Theidentity operator commutes withany other operator. This is the reason that the direction, not the length, of a vector describes the state of a system. Eigenvalues(observable values) not changed by length of vectoras long as . Some important theorems and proofs Copyright – Michael D. Fayer, 2007

29. Several independent eigenkets of an operator acan have the same eigenvalues. Degenerate Any superposition of these eigenkets is an eigenketwith the same eigenvalue. Example: px, py, and pz orbitals of the H atom. Copyright – Michael D. Fayer, 2007

30. Assume not the same. Hermitian property Left multiply top byRight multiply bottom by Subtract But, Theory is symmetrical in kets and bras. Therefore, a = b The eigenvalues associated with the eigenkets are the same as those for the eigenbras for the same linear operator. Copyright – Michael D. Fayer, 2007

31. take complex conjugate Hermitian property Left multiply the first equation byRight multiply third equation by Subtract Length of vector greater than zero. Therefore, ; and a number equal to its complex conjugate is real. Observables are real numbers. The eigenvalues of Hermitian operators are real. Copyright – Michael D. Fayer, 2007

32. The Orthogonality Theorem Two eigenvectors of a real dynamical variable (Hermitian operator - observable) belonging to different eigenvalues are orthogonal. (1) (2) Take complex conjugate of (1). Operator Hermitian; eigenvalue real Right multiply by (3) Left multiply (2) by (4) Subtract (4) from (3) But different eigenvalues Therefore, and are orthogonal. Copyright – Michael D. Fayer, 2007