04. Five Principles of Quantum Mechanics

Transcription

1 04. Five Principles of Quantum Mechanics () States are represented by vectors of length. A physical system is represented by a linear vector space (the space of all its possible states). () Properties are represented by operators. So: An operator O represents a property. Let its eigenvectors λ represent the value states ("eigenstates") associated with the property. Let its eigenvalues λ represent the (numerical) values of the property. Eigenvector/Eigenvalue (EE) Rule: A state possesses the value λ of a property represented by operator O if and only if that state is an eigenvector of O with eigenvalue λ.

2 Why is this helpful? Recall: White electrons appear to have no determinate value of Hardness. Let's represent the values of Hardness and Color as orthonormal basis vectors. Let's suppose the Hardness basis { hard, soft } is rotated by 45 with respect to the Color basis { white, black }: soft Then: white = hard + soft black 45 white hard An electron in a white state is in a "superposition" of hard and soft states. So: Since an electron in the state white cannot be in either of the states hard, soft, it cannot be said to possess a value of Hardness.

3 Let's be a bit more precise... Represent the Hardness basis vectors by column vectors: hard = soft = 0 0 Define the Hardness operator by H = 0 0. So: hard and soft are eigenvectors of H: Orthonormality check: hard soft = (, 0) 0 = 0 hard hard = (, 0 ) 0 = soft soft = ( 0, ) 0 = H hard = = 0 = + hard H soft = = 0 = soft Stipulate: + is the number corresponding to the Hardness value hard. is the number corresponding to the Hardness value soft. Thus: An electron in the state hard has a Hardness value of hard. An electron in the state soft has a Hardness value of soft.

4 Represent the Color basis vectors by column vectors: black = white = Note: The angle between black and soft is 45 black soft =, ( ) 0 = = cos 45 Define the Color operator by C = 0 0. So: black and white are eigenvectors of C: C black = 0 0 = + black C white = 0 0 Orthonormality check: ( ) black white =, ( ) black black =, white white =, Stipulate: + is the number corresponding to the Color value black. is the number corresponding to the Color value white. = white ( ) = 0 = =

5 The EE Rule says: To say a white electron has a Hardness value (hard or soft), it must be in an eigenstate of the Hardness operator. So: According to the Eigenvector/Eigenvalue Rule... - A white electron has no definite value of Hardness. - A black electron has no definite value of Hardness. - A hard/soft electron has no definite value of Color. Can now expand Color states in Hardness basis: black = = = hard + soft white = = 0 0 = hard soft But: The state represented by white is not an eigenstate of the operator H representing the Hardness property: H white = 0 0 = λ white, for any value of λ.

6 (3) Dynamics: States evolve in time via the Schrödinger equation. Plug an initial state ψ(t ) into the Schrödinger equation, and it produces a unique final state ψ(t ). state at time t ψ(t ) ψ(t ) Schrödinger evolution state at later time t The Schrödinger equation can be encoded in an operator S (which is a function of the Hamiltonian operator H that encodes the energy of the system). e iht /! A S A = A' state at time t state at later time t Important property: S is a linear operator. S(α A + β B ) = αs A + βs B, where α, β are numbers.

7 Recall: Experimental Result #: There is no correlation between Hardness measurements and Color measurements. - If the Hardness of a batch of white electrons is measured, 50% will be soft and 50% will be hard. Let's assume: "Born Rule": The probability that a quantum system in a state ψ possesses the value b of a property B is given by the square of the expansion coefficient of the basis state b in the expansion of ψ in the basis corresponding to all values of the property. Max Born (88-970) So: The probability that a white electron has the value hard when measured for Hardness is /! white = hard + soft An electron in a white state has a probability of / of being hard upon measurement for Hardness. More precisely...

8 (4) Born Rule. The probability that a state ψ possesses the value b i of the property represented by B is given by Pr(value of B is b i in state ψ ) ψ b i = a i where b i is the eigenvector of B with eigenvalue b i, and a i is the expansion coefficient corresponding to b i in the expansion of ψ in the eigenvector basis of B. Suppose a physical system is in a state represented by ψ. To measure the value of a property represented by an operator B: () First expand ψ in a basis given by a set of eigenvectors of B: ψ = a b + a b a N b N eigenvectors of B: B b = b b, B b = b b, etc... expansion coefficients of ψ in basis b,..., b N () The probability that ψ possesses the value b, say, of the property represented by B is then a, according to the Born Rule.

9 (4) Born Rule. The probability that a state ψ possesses the value b i of the property represented by B is given by Pr(value of B is b i in state ψ ) ψ b i = a i where b i is the eigenvector of B with eigenvalue b i, and a i is the expansion coefficient corresponding to b i in the expansion of ψ in the eigenvector basis of B. When ψ is itself an eigenvector b i of B, then the probability that it possesses the value b i is equal to. If: ψ = b i. Then: ψ b i = b i b i =. This is consistent with the EE Rule!

10 (5) Projection Postulate. When a measurement of a property B is made on a system in the state ψ = a b a N b N expanded in the eigenvector basis of B, and the result is the value b i, then ψ collapses to the state b i : ψ b i. collapse John von Neumann ( ) Motivation: Guarantees that if we obtain the value b i once, then we should get the same value b i on a second measurement (provided the system is not interferred with in-between). Check: - Suppose we conduct a B-measurement on the state ψ = a b a N b N. - Born Rule says: The probability of getting b i is a i <. - Suppose we get b i upon initial measurement. - Projection Postulate says: ψ collapses to b i. - Born Rule says: The probability of getting b i upon a second measurement is! - So: If we measure the property represented by B again, we should get b i with certainty.

11 The Wave Function Have been considering Color and Hardness (i.e., spin) properties for electrons: -Only values. -State space is a -dimensional vector space. -Many orthonormal bases; each associated with a spin property: Color, Hardness, Gelb, Scrad, etc.; all of which are mutually incompatible. Now consider Position and Momentum properties: -Infinite continuum of values. -State space is an infinite-dimensional vector space. -Two distinct orthonormal bases; one associated with Position, the other with Momentum; both of which are mutually incompatible. Let X be the operator that represents the property of Position. Let the eigenvectors and eigenvalues of X be given by x and x: X x = x x There are an infinite number of values for x, and thus an infinite number of eigenvectors x!

12 Suppose ψ represents the state of an electron located at some position. Can expand ψ in eigenvectors x of position operator X: ψ = a + a a All of the infinite number of position eigenvectors x are orthogonal to each other and form a basis for the -dim coordinate state space. Any expansion coefficient a x is given by a x = ψ x, where x can be any number from to +! ψ x is a continuous function, call it ψ(x), of x called the wave function. According to the Born Rule: Pr(electron is located at position x in state ψ ) = ψ x = ψ(x)