05. Multiparticle Systems

Transcription

1 05. Multiparticle Systems I. -Particle Product Spaces Suppose: Particle and particle are represented by vector spaces V and W. Then: The composite -particle system is represented by a product vector space V W. Let V and W be n-dim and m-dim vector spaces. The product vector space V W is an (n m)-dim vector space with the following property: For any v V, w W, one can form a vector ψ V W via the "tensor product" v w, which satisfies: (i) ( v + v ) w = v w + v w (ii) v ( w + w ) = v w + v w (iii) α( v w ) = α v w = v α w, for any scalar α. Instead of " v w ", we can alternatively write " v w " or " vw ".

2 Further characteristics of the -particle product space V W. An inner-product on V W is defined by the following: For any ψ = vw, ϕ = tu V W, with v, t V and w, u W, ψ ϕ v t w u. If { v, v,..., v n } and { w, w,..., w m } are bases for V and W, then a basis for V W is given by { v w, v w,..., v w m, v w,..., v n w m } 3. Any vector ψ in V W can be expanded in this basis: ψ = a v w + a v w a v w a nm v n w m 4. Let A and B be operators on V and W such that A v = a v, B w = b w, where v V, w W. Then there is an operator A B on V W such that (A B) vw = ab vw

3 Extension to multiparticle (multi-partite) systems A product vector space H may be formed from the tensor product of more than two lower-dim vector spaces. A tensor product space H may admit more than one decomposition into lower-dim vector spaces. Ex. Let H be a 6-dim vector space. Then: One can always find -dim vector spaces V, V, V 3, V 4 such that H = V V V 3 V 4 And: One can always find 4-dim vector spaces W, W such that H = W W Note: A "factor" vector space must have dim >. So: As long as the dimension n of a vector space isn't a prime number, it will admit at least one decomposition into the tensor product of lower-dim vector spaces. And: How many it will admit depends on the prime factorization of n.

4 -particle example Consider -dim spin state spaces V, W for two electrons. Combined -particle spin space is given by 4-dim V W. { hard, soft } is a basis for V and { hard, soft } is a basis for W. { hard hard, hard soft, soft hard, soft soft } is a basis for V W. Any -particle state A in V W can be expanded in this basis: A = a hard hard + a hard soft + a soft hard + a soft soft II. Entangled States An entangled state in a product vector space H with respect to a decomposition H = V! V n is a vector ψ that cannot be written as a product of n terms, ψ = v! v n, where v i V i. Erwin Schrödinger Note: Entanglement is a relative property! A vector in H can be entangled with respect to one decomposition of H, but not entangled with respect to another decomposition of H.

5 Examples: Entangled: Nonentangled (Separable): A = 4 = 4 B = According to the Eigenvalue-Eigenvector Rule: In states Ψ + and A, both electrons have no determinate Hardness value, but the combined system as a whole does have a determinate value of some other property.* In state B, electron has no determinate Hardness value, but electron does (i.e., hard), and the combined system as a whole has a determinate value of some other property.* { hard hard + hard soft + soft hard + soft soft } { hard + soft }{ hard + soft } { hard hard + soft hard } = C = hard hard { } Ψ + = hard hard + soft soft In state C, both electrons have determinate Hardness values, and the combined system as a whole has a determinate value of some other property.* *General fact: Any vector is the eigenvector of some operator. { hard + soft } hard

6 So: According to the EE Rule, in any -particle state, either particle may or may not have well-defined properties. But: According to EE, the combined -particle system as a whole will always have well-defined properties! Why? Because, again, any vector in a vector space (including V W) is an eigenvector of some (Hermitian) operator on that space. So there exist - particle operators with eigenvectors Ψ +, A, B and C that represent properties of the -particle system as a whole.

7 -Particle "Holistic" Properties Suppose: Q = { } eigenvector of particle position operator X () eigenvector of particle position operator X () If we only want to measure P's position (and not P's), we must use the - particle operator X () I () (I () = identity operator on W). If we only want to measure P's position (and not P's), we must use the - particle operator I () X () (I () = identity operator on V). The difference in the positions of P and P is a property of the -particle system represented by the -particle operator (I () X () ) (X () I () ). Claim: Q is an eigenstate of (I () X () ) (X () I () ) but not of X () I () or I () X ()! So: According to the EE Rule, particles and have no definite position in the -particle state Q, but the difference in their positions is a definite property of the -particle state as a whole!

8 Claim: Q is an eigenstate of (I () X () ) (X () I () ), but not of X () I () or I () X (). Check: (a) Q is an eigenstate of (I X () ) (X I () ): {(I () X () ) (X () I () )} Q = (I () X () ) Q (X () I () ) Q = (I () X () ) 5 { } (X () I () ) 5 { } = 7 5 { } 5 5 { } = 5 { } = Q (b) Q is not an eigenstate of X () I () or I () X () : { } { } (X () I () ) Q = (X () I () ) = λ Q, for any value of λ. In the state represented by Q, the value of the difference-in-position operator is ; i.e., Particle and Particle differ in position by. Similarly for I () X ().

9 III. Born Rule for -Particle States Suppose: A -particle system is in a state represented by k, and A () and B () are operators that represent properties of P and P. Then: The probability that the value of A () is a i and the value of B () is b i is: Pr(value of A () is a i and value of B () is b i in state k ) a i b i k Where: a i b i (or a i b i ) is an eigenvector of the -particle operator A () B (). And: The probability that the value of A () is a i is: Pr(value of A () is a i in state k ) a i l k + a i l k a i l N k Where: a i l j, j =,..., N, are eigenvectors of the -particle operator A () L (), for any arbitrary operator L (). Motivation (Law of Total Probability): The probability that the value of A () is a i is equal to the sum of the probabilities of all the different ways in which the value of A () could be a i.

10 IV. -Particle Projection Postulate Suppose: A -particle system is in a state represented by D, and a property of P represented by A () is measured with the resulting value a i. Then: D collapses to the state given by the following: (a) Expand D in eigenvectors of the -particle operator A () L (), for any arbitrary operator L () : D = d a l d N a l N + d a l d NN a N l N (b) Throw out all terms other than ones with a i. Then divide by an appropriate normalization term Λ to make sure the result is a vector with unit length: D collapse d i a i l + d i a i l +... Λ

11 Example : Suppose: D = q 3 m 4 is an eigenvector of Q () M (). Now: Suppose the property represented by A () is measured with the resulting value a 5. What happens to D? First: Expand D in the eigenvectors of A () L (), for any arbitrary L (). Note: The P part of D already is an eigenvector of M (). So: Use eigenvectors of A () M () for simplicity. D = q 3 m 4 = d a m 4 + d a m d N a N m 4 Next: Throw out all terms other than ones with a 5, and normalize the result. This just leaves d 5 a 5 m 4. To normalize it, divide by its length, which is just d 5. So: D collapse a 5 m 4

12 Example (collapse of entangled state): { } Suppose: D = a 4 l 7 + a 5 l 4 Now: Suppose the property represented by A () is measured with the resulting value a 5. What happens to D? Note: The P part of D is already in an eigenvector basis of A (). So: Simply throw out all terms in D that don't contain a 5. Result: D = a 5 l 4 So: D a collapse 5 l 4 Important Note: The collapse of D upon measurement of a property of P automatically changes the state of the unmeasured P! More on this to come.

13 V. -Path Experiment Again. Without Barrier: y 4 y 3 t 3 t 4 00% of exiting electrons are white. y t hard t 3 y Color white t Hardness soft t x x x 3 x 4 x 5 At t, the electron's state is: white x,y = hard x {,y + soft x,y } At t, the electron's state is: At t 3, the electron's state is: { hard x,y + soft x 3,y } { hard x 3,y 3 + soft x 4,y } entangled states! At t 4, the electron's state is: { hard x 5,y 4 + soft x 5,y 4 } = white x 5,y 4 Pr(value of C is white in state at t 4 ) = white, x 5, y 4 white, x 5, y 4 =

14 V. -Path Experiment Again. With Barrier: y 4 y 3 t 3 t 4 50% of exiting electrons are white, 50% are black. y t hard t 3 y Color white t Hardness soft x x x 3 x 4 x 5 At t 4, the electron's state is: k = To measure Color at t 4, expand k in Color basis: k = { } x 5,y 4 + ( )( ) black + white { hard x 5,y 4 + soft x 3,y } ( ) { } x 3,y ( ) black white = black x 5,y 4 + black x 3,y + white x 5,y 4 white x 3,y Pr(value of C is white in state k ) = white, x 5, y 4 k + white, x 3, y k = + =