Quantum Mechanics I Physics 5701

Transcription

1 Quantum Mechanics I Physics 5701 Z. E. Meziani 02/23//2017 Physics 5701 Lecture

2 Outline 1 General Formulation of Quantum Mechanics 2 Measurement of physical quantities and observables 3 Representations in specific bases. Wave function. 4 Structure of the Hilbert space. Tensor product of spaces

3 General Formulation of Quantum Mechanics 1 States and principle of superposition

4 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables

5 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification

6 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition

7 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation

8 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation 4 Representation in particular bases; wave functions

9 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation 4 Representation in particular bases; wave functions 5 Structure of the Hilbert space, Tensor product of spaces

10 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation 4 Representation in particular bases; wave functions 5 Structure of the Hilbert space, Tensor product of spaces We will describe, the state of a system

11 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation 4 Representation in particular bases; wave functions 5 Structure of the Hilbert space, Tensor product of spaces We will describe, the state of a system The measurement of a physical quantity

12 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation 4 Representation in particular bases; wave functions 5 Structure of the Hilbert space, Tensor product of spaces We will describe, the state of a system The measurement of a physical quantity The evolution of a system

13 General Formulation of Quantum Mechanics 1 States and principle of superposition 2 Measurement of physical quantities and observables Principle of quantification Spectral decomposition 3 Dynamical equation 4 Representation in particular bases; wave functions 5 Structure of the Hilbert space, Tensor product of spaces We will describe, the state of a system The measurement of a physical quantity The evolution of a system Remark: The described principles are relative to systems that are in "pure states". The notion of "statistical mixing"will be described later. Example: linearly or circularly polarized light is in a "pure state" in contrast to unpolarized or partially polarized light.

14 Postulate I: Principle of Superposition The state of a system is totally defined, at each time, by an element of the appropriate Hilbert space E ψ(t) Since any linear superposition of states is also a state vector we write ψ = n c i ψ i c i C and ψ i E i=1

15 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable a) To each physical quantity A is associated a Hermitian operator  acting on E.  is the observable representing the quantity A.

16 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable a) To each physical quantity A is associated a Hermitian operator  acting on E.  is the observable representing the quantity A. b) Assume the system is in state ψ at the time of the measurement of the quantity A. For any ψ the only possible results of the measurement are the eigenvalues a n of the observable Â.

17 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable a) To each physical quantity A is associated a Hermitian operator  acting on E.  is the observable representing the quantity A. b) Assume the system is in state ψ at the time of the measurement of the quantity A. For any ψ the only possible results of the measurement are the eigenvalues a n of the observable Â. c) When the quantity A is measured on a system in the normalized state ψ the probability P(a n) of obtaining the non-degenerate eigenvalues a n of the corresponding observable  is P(a n) = u n ψ 2 where u n is the normalized eigenvector of  associated with the eigenvalue a n.

18 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable a) To each physical quantity A is associated a Hermitian operator  acting on E.  is the observable representing the quantity A. b) Assume the system is in state ψ at the time of the measurement of the quantity A. For any ψ the only possible results of the measurement are the eigenvalues a n of the observable Â. c) When the quantity A is measured on a system in the normalized state ψ the probability P(a n) of obtaining the non-degenerate eigenvalues a n of the corresponding observable  is P(a n) = u n ψ 2 where u n is the normalized eigenvector of  associated with the eigenvalue a n. for a degenerate spectrum P(a n) = g n i u n ψ 2 { u i n } with i = 1, 2..., g n is a basis in subspace E n with

19 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable (Continued) d) The state of a system immediately after a measurement which gave a n is the normalized projection P n ψ ψ Pn ψ of ψ onto the eigensubspace associated with a n. ψ represent the system immediately before the measurement.

20 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable (Continued) d) The state of a system immediately after a measurement which gave a n is the normalized projection P n ψ ψ Pn ψ of ψ onto the eigensubspace associated with a n. ψ represent the system immediately before the measurement. In c) and d) we assumed that ψ is normalized to unity, (ψ ) = 1. It is our way to distinguish between ψ and c ψ with c C since they lead to the same physical result. The convention, however, still has an ambiguity for phase factors e iδ where δ is real. If ψ 1 = eiδ 1 ψ 1 and ψ 2 = eiδ 2 ψ 2 the state superposition c 1 ψ 1 + c2 ψ 2 is different from c1 ψ1 + c2 ψ2

21 Measurement of physical quantities and observables Postulate II: Measurement of a physical quantity and observable (Continued) d) The state of a system immediately after a measurement which gave a n is the normalized projection P n ψ ψ Pn ψ of ψ onto the eigensubspace associated with a n. ψ represent the system immediately before the measurement. In c) and d) we assumed that ψ is normalized to unity, (ψ ) = 1. It is our way to distinguish between ψ and c ψ with c C since they lead to the same physical result. The convention, however, still has an ambiguity for phase factors e iδ where δ is real. If ψ 1 = eiδ 1 ψ 1 and ψ 2 = eiδ 2 ψ 2 the state superposition c 1 ψ 1 + c2 ψ 2 is different from c1 ψ1 + c2 ψ2 When we study a particular system we cannot modify as we wish the relative phases of the different vectors of the system.

22 Measurement of physical quantities and observables Terminology II b) is called the "principle of quantization"since the spectrum is discrete the possible values of the physical quantity A are discrete.

23 Measurement of physical quantities and observables Terminology II b) is called the "principle of quantization"since the spectrum is discrete the possible values of the physical quantity A are discrete. II c) is called the "principle of spectral decomposition"since u n ψ are the probability amplitudes that tell us whether the system is in u 1, u 2 or u n during a measurement when the system is in state ψ.

24 Measurement of physical quantities and observables Terminology II b) is called the "principle of quantization"since the spectrum is discrete the possible values of the physical quantity A are discrete. II c) is called the "principle of spectral decomposition"since u n ψ are the probability amplitudes that tell us whether the system is in u 1, u 2 or u n during a measurement when the system is in state ψ. II d) is called "Reduction of a wave packet". Just after the measurement the system is in state u 2 since the measurement gave a 2.

25 Measurement of physical quantities and observables Continuous case When the physical quantity A is measured on a system in the normalized state ψ the probability dp(α) of obtaining a result included between α and α + dα is equal to dp(α) = v α ψ 2 dα where v α is the eigenvector corresponding to the eigenvalue α of the observable  associated with A.

26 Measurement of physical quantities and observables Mean value of a measurement; Mean value of an observable in a given state Knowing the probability dp(a n) of finding a n after a measurement of A one can compute the average value a ψ = n a np(a n) where P(a n) = ψ P n ψ a ψ = n a n ψ P n ψ a n ˆPn = g n a n u i n u i n = g n i=1 i=1 a ψ = ψ  ˆP n ψ n  u i n u i n = Ân ˆP n but since the closure relation reads n ˆP n = ˆ1, then a ψ = ψ  ψ

27 Measurement of physical quantities and observables Dynamical Equation: Postulate III Let ψ(t) be the state of the system at time t. As long as the system is not under observation, it s time evolution is governed by the equation i d ψ(t) = Ĥ ψ(t) dt where Ĥ is the observable associated to the energy of the system of the Hamiltonian of the system. This is the generalized form of Schrödinger equation.

28 Measurement of physical quantities and observables Dynamical Equation: Postulate III Let ψ(t) be the state of the system at time t. As long as the system is not under observation, it s time evolution is governed by the equation i d ψ(t) = Ĥ ψ(t) dt where Ĥ is the observable associated to the energy of the system of the Hamiltonian of the system. This is the generalized form of Schrödinger equation. Consequences:

29 Measurement of physical quantities and observables Dynamical Equation: Postulate III Let ψ(t) be the state of the system at time t. As long as the system is not under observation, it s time evolution is governed by the equation i d ψ(t) = Ĥ ψ(t) dt where Ĥ is the observable associated to the energy of the system of the Hamiltonian of the system. This is the generalized form of Schrödinger equation. Consequences: i) Conservation of the norm: First we can show that the norm of ψ is constant with time, this is a condition necessary for the coherence of the theory. We need to use the third postulate and the Hermitian property of Ĥ. Ĥ = Ĥ since Ĥ is associated to a physical quantity (the energy). i d dt ψ = Ĥ ψ i ψ ( d dt ψ ) = ψ Ĥ ψ i d dt ψ = ψ Ĥ i ( d dt ψ ) ψ = ψ Ĥψ After subtraction we obtain Thus the norm is conserved in time. d dt ψ ψ = 0

30 Measurement of physical quantities and observables Dynamical Equation: Postulate III Let ψ(t) be the state of the system at time t. As long as the system is not under observation, it s time evolution is governed by the equation i d ψ(t) = Ĥ ψ(t) dt where Ĥ is the observable associated to the energy of the system of the Hamiltonian of the system. This is the generalized form of Schrödinger equation. Consequences: i) Conservation of the norm: First we can show that the norm of ψ is constant with time, this is a condition necessary for the coherence of the theory. We need to use the third postulate and the Hermitian property of Ĥ. Ĥ = Ĥ since Ĥ is associated to a physical quantity (the energy). i d dt ψ = Ĥ ψ i ψ ( d dt ψ ) = ψ Ĥ ψ i d dt ψ = ψ Ĥ i ( d dt ψ ) ψ = ψ Ĥψ After subtraction we obtain d dt ψ ψ = 0 Thus the norm is conserved in time. ii) Eigenstates of the energy: Time dependence of a state vector

31 Measurement of physical quantities and observables Eigenstates of the energy: Time dependence of a state vector The eigenstates of the energy correspond to the eigenvectors of Ĥ Ĥ ψ n = E n ψ n Assume E n non-degenerate thus the subspaces associated with these eigenvalues are of dimension one. ψ n constitute a basis of Eallowing to expand any vector state ψ. ψ(t = 0) = n c n ψ n with c n = ψ n ψ(t = 0) at time t ψ(t) = λ(t) ψ n with λ n(0) = c n n

32 Measurement of physical quantities and observables Eigenstates of the energy: Time dependence of a state vector The eigenstates of the energy correspond to the eigenvectors of Ĥ Ĥ ψ n = E n ψ n Assume E n non-degenerate thus the subspaces associated with these eigenvalues are of dimension one. ψ n constitute a basis of Eallowing to expand any vector state ψ. ψ(t = 0) = n c n ψ n with c n = ψ n ψ(t = 0) at time t ψ(t) = n λ(t) ψ n with λ n(0) = c n The Schrödinger equation gives: i d dt λn(t) ψn = n n Since ψ n are orthonormal, then λ n(t)e n ψ n i d λn(t) = Enλn(t) λn(t) = cne ient/ dt ψ(t) = c nλ n(t) = c ne ient/ ψ n

33 Representations in specific bases. Wave function. Representations in specific bases. Wave function. Assume a basis { k } of Hilbert space, each ket ψ(t) can be represented by its coordinates {c k (t)} c k (t) = k ψ(t) This leads to the matrix representation of states and operators of Hilbert space. We will see that it is always nice to work in the eigenbasis of this or that observable.

34 Representations in specific bases. Wave function. Representations in specific bases. Wave function. Assume a basis { k } of Hilbert space, each ket ψ(t) can be represented by its coordinates {c k (t)} c k (t) = k ψ(t) This leads to the matrix representation of states and operators of Hilbert space. We will see that it is always nice to work in the eigenbasis of this or that observable. Some observables, however, in the case of infinite dimension space have "eigenvectors" that do not belong to the Hilbert space. This happens when all or just part of their spectrum is continuous. This is the case of the R and P observables.

35 Representations in specific bases. Wave function. Representations in specific bases. Wave function. Assume a basis { k } of Hilbert space, each ket ψ(t) can be represented by its coordinates {c k (t)} c k (t) = k ψ(t) This leads to the matrix representation of states and operators of Hilbert space. We will see that it is always nice to work in the eigenbasis of this or that observable. Some observables, however, in the case of infinite dimension space have "eigenvectors" that do not belong to the Hilbert space. This happens when all or just part of their spectrum is continuous. This is the case of the R and P observables. r and p do not belong to E but to a larger space.

36 Representations in specific bases. Wave function. Representations in specific bases. Wave function. Assume a basis { k } of Hilbert space, each ket ψ(t) can be represented by its coordinates {c k (t)} c k (t) = k ψ(t) This leads to the matrix representation of states and operators of Hilbert space. We will see that it is always nice to work in the eigenbasis of this or that observable. Some observables, however, in the case of infinite dimension space have "eigenvectors" that do not belong to the Hilbert space. This happens when all or just part of their spectrum is continuous. This is the case of the R and P observables. r and p do not belong to E but to a larger space. The orthonormalization relation for a continuous basis { v α } will be δ αα = α α x x = δ(x x )

37 Representations in specific bases. Wave function. Representations in specific bases. Wave function (continued) Closure Relation: α α = 1 n dx x x

38 Representations in specific bases. Wave function. Representations in specific bases. Wave function (continued) Closure Relation: α α = 1 dx x x n ψ(x) = x ψ are the coefficients of ψ in the basis { x >}

39 Representations in specific bases. Wave function. Representations in specific bases. Wave function (continued) Closure Relation: α α = 1 dx x x n ψ(x) = x ψ are the coefficients of ψ in the basis { x >} Probability of finding a α becomes dp(x) = ψ (x)ψ(x)dx

40 Representations in specific bases. Wave function. Representations in specific bases. Wave function (continued) Closure Relation: α α = 1 dx x x n ψ(x) = x ψ are the coefficients of ψ in the basis { x >} Probability of finding a α becomes dp(x) = ψ (x)ψ(x)dx For a particle in space the wave function is obviously ψ( r, t) = r ψ(t) and its Fourier transform φ( p, t) = p ψ(t)

41 Representations in specific bases. Wave function. Representations in specific bases. Wave function (continued) Closure Relation: α α = 1 n dx x x ψ(x) = x ψ are the coefficients of ψ in the basis { x >} Probability of finding a α becomes dp(x) = ψ (x)ψ(x)dx For a particle in space the wave function is obviously and its Fourier transform ψ( r, t) = r ψ(t) φ( p, t) = p ψ(t) The matrix for changing basis from { r } to { p } is r p = 1 p r (2π ) 3 /2 ei (1) it is also the "wave function" corresponding to an "eigenstate" of ˆ P

42 Structure of the Hilbert space. Tensor product of spaces Structure of the Hilbert space. Tensor product of spaces What is the structure of the Hilbert space in which we describe a given system? For a free particle the motion is described in the L 2 Hilbert space (square integrable functions).

43 Structure of the Hilbert space. Tensor product of spaces Structure of the Hilbert space. Tensor product of spaces What is the structure of the Hilbert space in which we describe a given system? For a free particle the motion is described in the L 2 Hilbert space (square integrable functions). The intrinsic magnetic moment is described in a two dimensional Hilbert space.

44 Structure of the Hilbert space. Tensor product of spaces Structure of the Hilbert space. Tensor product of spaces What is the structure of the Hilbert space in which we describe a given system? For a free particle the motion is described in the L 2 Hilbert space (square integrable functions). The intrinsic magnetic moment is described in a two dimensional Hilbert space. We introduce the notion of "degrees of freedom" of a system. A particle in space has three degrees of freedom. Translation in each direction (Ox, Oy, Oz). If the particle has an intrinsic magnetic moment it has a degree of freedom associated with it, etc...

45 Structure of the Hilbert space. Tensor product of spaces Structure of the Hilbert space. Tensor product of spaces What is the structure of the Hilbert space in which we describe a given system? For a free particle the motion is described in the L 2 Hilbert space (square integrable functions). The intrinsic magnetic moment is described in a two dimensional Hilbert space. We introduce the notion of "degrees of freedom" of a system. A particle in space has three degrees of freedom. Translation in each direction (Ox, Oy, Oz). If the particle has an intrinsic magnetic moment it has a degree of freedom associated with it, etc... Each degree of freedom is described by a given Hilbert space.

46 Structure of the Hilbert space. Tensor product of spaces Structure of the Hilbert space. Tensor product of spaces What is the structure of the Hilbert space in which we describe a given system? For a free particle the motion is described in the L 2 Hilbert space (square integrable functions). The intrinsic magnetic moment is described in a two dimensional Hilbert space. We introduce the notion of "degrees of freedom" of a system. A particle in space has three degrees of freedom. Translation in each direction (Ox, Oy, Oz). If the particle has an intrinsic magnetic moment it has a degree of freedom associated with it, etc... Each degree of freedom is described by a given Hilbert space. We postulate that for a system of N degrees of freedom it is described in the Hilbert space "Tensor Product" of individual Hilbert spaces of the N degrees of freedom.

47 Structure of the Hilbert space. Tensor product of spaces Examples: 1 - Particle moving in 3 dimensions ψ n1,n 2,n 3 (x, y, z) = ψ n1 (x)ψ n2 (y)ψ n3 (z) ψ(x, y, z) = c n1,n 2,n 3 ψ n1,n 2,n 3 (x, y, z) n 1,n 2,n 3

48 Structure of the Hilbert space. Tensor product of spaces Examples: 1 - Particle moving in 3 dimensions ψ n1,n 2,n 3 (x, y, z) = ψ n1 (x)ψ n2 (y)ψ n3 (z) ψ(x, y, z) = c n1,n 2,n 3 ψ n1,n 2,n 3 (x, y, z) n 1,n 2,n System of 2 particles: ψ( r 1, r 2 ) = c n1,n 2 ψ n1 ( r 1 )ψ n2 ( r 2 ) n 1,n 2 Tensor product of space r 1 space and r 2 space

49 Structure of the Hilbert space. Tensor product of spaces Examples: 1 - Particle moving in 3 dimensions 2 - System of 2 particles: ψ n1,n 2,n 3 (x, y, z) = ψ n1 (x)ψ n2 (y)ψ n3 (z) ψ(x, y, z) = c n1,n 2,n 3 ψ n1,n 2,n 3 (x, y, z) n 1,n 2,n 3 ψ( r 1, r 2 ) = n 1,n 2 c n1,n 2 ψ n1 ( r 1 )ψ n2 ( r 2 ) Tensor product of space r 1 space and r 2 space 3 - To study the motion of a particle and its magnetic state: The Hilbert space is the Tensor product of L 2 and a Hilbert space of 2 dimensions. ( ) ψ+ ( r, t) ψ ( r, t) Probability to find the particle around r within d r with µ z = +µ 0 is given by ψ + ( r, t) 2 d r, same for ψ and µ z = µ 0.Probability to find µ z = +µ 0 is ψ + ( r, t) 2 d r