* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Lesson 5
Scalar field theory wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum entanglement wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum key distribution wikipedia , lookup
Self-adjoint operator wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Coherent states wikipedia , lookup
Ensemble interpretation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Compact operator on Hilbert space wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
EPR paradox wikipedia , lookup
Path integral formulation wikipedia , lookup
Canonical quantization wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Wave function wikipedia , lookup
Quantum group wikipedia , lookup
Hidden variable theory wikipedia , lookup
Dirac equation wikipedia , lookup
Density matrix wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum state wikipedia , lookup
Lesson 7 I. Dirac Delta Function - (x) The Dirac delta function is not actually a function, but a distribution (see either Arfkin or Butkov) which means that it is defined in terms of its integral properties. The two integral properties that define the Dirac delta function are 1 x x dx 0 b 1) o a if x o in the region between a and b otherwise f x if x in the region between a and b f x x x dx 0 otherwise b 2) o o o a EXAMPLE: Evaluate the integral e x 2 nπ x sin x a dx L 5 EXAMPLE: Evaluate 4 8 3x 2 y z 3 δx 1 δy 2 δz 3 dx dy dz x 0 y 0 z 5 II. Kroniker Delta Function - m,n 1 if m n 0 if m n m, n Example: Orthogonality and Normalization 2 mππ nππ sin sin dx δ m, n L L L III. Expansion Postulate For an arbitrary observable Ô with eigenvectors Φ n (x) and real eigenvalues on, the eigenfunctions form an orthogonal set that can be normalized so that m n dx m, n Furthermore, the set of eigenfunctions span the space so the wave function describing any quantum state can be expressed as a linear combination of these eigenstates as x c n n x n This fact leads us to consider the similarity between our work with eigenfunctions and our previous work in physics with unit vectors. IV. Dirac Notation (Bra-ket Notation) We will consider Dirac notation as a sort of vector shorthand for doing algebra in quantum mechanics but it is actually much more powerful and profound. A. B. ket 1. Symbol - Ψ 2. Ψ is interpreted as a vector representing . Bra 1. 2. Symbol - is interpreted as a vector representing . C. Inner Product 1. Symbol - 2. is defined as dx 3. is interpreted as giving the projection of the vector onto the vector . Special Cases 1. Orthogonality Two vectors are said to be orthogonal if their inner product is zero. 0 2. Normalization A vector is normalized if the inner product of the vector with itself is equal to 1. 1 D. Expansion Postulate We can now write the expansion postulate in terms of Dirac notation as cn n n where is the state vector representing the quantum system and n is the nth eigenvector for the operator Ô . The coefficients may be real or complex. In Dirac notation, eigenvectors act like unit vectors. Each operator has a set of eigenvectors which one can use to represent the state vector of the quantum system. Thus, you may represent the state vector of a quantum system as a combination of many different sets of eigenvectors in the same way that we can represent the position of a classical particle in different coordinate systems (cylindrical, Cartesian, spherical, etc.). However, in quantum mechanics the measurement process and its connection to the eigenvectors has a deeper importance as we shall see shortly. E. Normalization of a State Vector We know that the probabilistic interpretation of quantum mechanics requires that dx 1 If we now write the expansion of the bra and the ket in terms of a set of eigenvectors of some operator, we have cn n n c m m m Thus, our normalization requirement becomes c m c n m n m n c m c n m, n m n c n c n c n 1 2 n F. n Operators and Eigenvalues When an operator is applied to a ket that is one its eigenvectors, we obtain the ket times the associated eigenvalue.  n  n a n n G. Expectation Values The expectation value is the average value that one would obtain if one made a specific measurement of a large number of identical quantum systems (ensemble). Each measurement would produce an eigenvalue for the physical quantity which may or may not be equal tothe expectation value and would leave the system in the eigenstate that corresponds to the measured eigenvalue of the operator associated with the physical quantity being measured. In Dirac notation, we can write the expectation value as A  dx  In our expansion representation, we find the following useful result for a discrete set of eigenvectors. A  cm c n a n m n m n A c m m c n a n m, n n A a n cn 2 n We see that expectation value is found by multiplying the eigenvalue 2 an by the probability of obtaining that eigenvalue, c n . H. Additional Computation Properties All of the properties of Dirac algebra are defined either by our previous definition of the inner product or due to physical requirements. 1. Constants can be pulled outside of a bra or ket using the following relationships: i) a a ii) a a Both of these statements indicate that the constant increases the length of the vector (multiplication of a vector by a scalar). Proof of (i) Let x a x a dx a dx a a by comparison of the two sides of the equation, we have that a a Q.E.D. 2. Proof: dx dx dx dx Q.E.D. 3. Addition Property The ket of the sum of two wave functions is the sum of the ket vectors of the individual wave functions. The bra of the sum of two wave functions is the sum of the bra vectors of the individual wave functions. Dirac Calculation Example: The chief quantum mechanic of the quantum artillery has told you that they have discovered a new operator Ĝ (gun operator). The eigenvectors and eigenvalues for the gun operator are n 2 g n 2n nth eigenvecto r nth eigenvalue You are also told that an atomic artillery unit at Ft. Small has a state vector given by K 2 1 3 2 3 A. Normalize the state vector B. What are the possible values that can be obtained for the guns in the artillery unit? C. What is the probability that we measure 8 guns? D. What would be the artillery state vector immediately following a measurement in which we found 18 guns? E. If we made a second gun measurement on the artillery system in D, what would we obtain for the number of guns? F. For the artillery unit in part B, what is the expectation value for the number of guns?