Kets, bras, and the operators

Transcription

1 Quantum mechanics I, lecture 2, September 20, 2012 L, Bolotov Univ. of Tsukuba Kets, bras, and the operators The textbooks: J.J. Sakurai, section ; N. Zettili,, sections , "Science is organized knowledge" (Herbert Spencer).

2 Outline Formulate the basic mathematics of vector spaces as used in quantum mechanics. o Five fundamental postulates; o The Hilbert vector space; o The Dirac s notation kets and bras spaces; o Observables as operators; o Matrix representation. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 2

3 Fundamental postulates of Quantum Mechanics State of any quantum system is specified by a state vector ψ in the Hilbert space. Observables and operators: to any physically measurable quantity A there corresponds a linear Hermitian operator A. Measurements and eigenvalues of operators: the measurement of A is represented by the action of the operator A on the state vector ψ, A ψ t = a n ψ n a n - the result of measurements, ψ n - the state after the measurements. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 3

4 Fundamental postulates of Quantum Mechanics Probabilistic outcome of measurements: (next lecture) when measuring A of the system in a state ψ, there is the probability of obtaining one of the eigenvalue a n, which is given by P n a n = ψ n ψ 2 ψ ψ = a n 2 ψ ψ Time evolution (later in the course) is governed by the Schrodinger equation, is given by (H- an operator of the total energy) ψ = H ψ t September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 4

5 A linear vector space A linear vector space is defined as Consists of two sets of elements: A set of vectors ψ, φ, χ and a set of complex scalars a, b, c, Two algebraic rules: A rule for vector addition: ψ + φ = χ ψ + φ = φ + χ O + ψ = ψ ψ + ψ = O and a rule for scalar multiplication: a ψ + φ = aψ + aφ, a + b ψ = aψ + bψ a b ψ = ab ψ I ψ = ψ I = ψ, o ψ = ψ o = o a unity scalar (I) and a zero scalar (o) ψ φ χ September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 5

6 The Hilbert space The Hilbert space is a linear vector space A scalar product is strictly positive φ, ψ = φ ψ =complex number φ, ψ = ψ, φ ψ, ψ = ψ ψ = ψ 2 0 ψ ψ 3 There exists a sequence ψ n For any ψ there exists ψ n that : lim n ψ ψ n = 0 (in other words, the sequence ψ n converges to ψ) ψ 1 ψ 2 September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 6

7 The Euclidean vector space k A The Euclidean vector space is a Hilbert space. The basis consists of 3 linearly independent vectors. The dimension of the Euclidean space is 3. j i For any vector A = a 1 i + a 2 j + a 3 k A = A the length, a 1 = i A, a 2 = j A, a 3 = k A For any vector C = A + B The basis vectors: i, j, k i j = k j = i k = 0 i = j = k = 1 A C B September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 7

8 Other examples of the Hilbert space The space of complex functions is a Hilbert space. The basis consists of independent basis functions. The dimension of the function space is (infinity) The real functions of x-axis f x = 4 g x = x 2 h x = e 2x The complex functions on t-axis f t = 5 + i t g t i π = ωt 2 h t = e iωt 1 ω 2 September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 8

9 The Dirac s notation The physical state of a system is represented in quantum mechanics by elements of a Hilbert space. The elements are called state vectors For example, a state vector S z + - a spin state of Ag atom - is the element of the Hilbert space. P.A.M. Dirac ( , an English theoretical physicist) introduces the concepts of kets, bras, and bra-kets, which provides a way to operate the state vectors with ease and clarity. Following Dirac, a KET is a state vector in a complex vector space, we denote it by α This state ket contains complete information about the physical state. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 9

10 KET properties Two kets can be added: α + β = γ A ket can be multiplied by a complex number (from right and left): c 1 α = α c 1 = γ Any arbitrary ket can be written as a linear combination of others α = n c n n It is the physics postulate that α and c 1 α represent the same physical state. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 10

11 BRA space Every vector space has an associated dual space. Following Dirac, a BRA is a vector in a bra-space dual to the ket-space. We denote it by α To every ket α there corresponds a unique bra α, and vice versa: β β There is a one-to-one correspondence between bras and kets : a α + b β a α + b β Common notations: a ψ = a ψ, aψ = a ψ Complex conjugate September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 11

12 Inner product The inner product (the scalar product) of a bra and a ket is written as β α = β α bra (c) ket Note: in forming an inner product, we always take one vector from the bra-space and one vector from the ket-space. We postulate two fundamental properties of inner product: 1. complex conjugate β α = α β 2. Positive definite metric: α α = α α 0 and a REAL number September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 12

13 Example Find: aφ + bψ cχ =? aφ + bψ cχ = aφ cχ + bψ cχ = = a c φ χ + b c ψ χ September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 13

14 Additional properties of inner product Inner product Two kets are said to be orthogonal if α β = 0 The state ket is said to be normalized when α α = 1 Scalar product in the Euclidean space Two vectors are to be orthogonal if r s = 0 The vector is normalized when r r = 1 We can form a normalized ket : 1 a = α α α where a a = 1 We call α α as the norm of α We can form a normalized vector : 1 n = r r r where n n = 1 We call r r = r as the norm of r September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 14

15 Products of the type Forbidden quantities φ ψ ket () ket φ ψ bra () bra are nonsensical, when φ and ψ belong to the same vector space. However, products φ ψ is meaningful ONLY when φ and ψ belong to different vector spaces (an orbital angular momentum space and a spin space). It appears in the angular momentum formalism. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 15

16 Analogy In wave mechanics, a state vector is given by a complex function of a function space (a Hilbert space). The inner product is defined by a finite integral over a volume ψ, φ = ψ r φ r dr For a wave function ψ r, t of a particle, the quantity ψ, ψ = ψ r, t 2 d 3 r represents the probability to find the particle in a volume at time t. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 16

17 Physical meaning of inner product 1. The product φ ψ represents the projection of ψ onto φ. (by analogy with the scalar product of vectors in the Euclidean space) 2. The quantity φ ψ represents the probability amplitude that the system in a state ψ will be found in another state φ after a measurements. (Here, both ψ and ψ are normalized states) September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 17

18 Summary 1 A physical state of a quantum system is represented by a state vector (a state KET) in a Hilbert space. There exists a dual space (BRA-space). There is a one-to-one correspondence between KET- and BRA- spaces. is defined as the bra-c-ket product (a complex number) β α = β α bra (c) ket Products of the type β α are forbidden when the kets are in the same space. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 18

19 Observables as operators An observable can be represented by an operator in the vector space in question. An operator is a mathematical rule. When an operator applied to a ket from the left, the ket is transformed into another ket of the same space. (A applied to a bra from the right another bra) A ψ = ψ, φ A = φ Examples: the unity (identity) operator applied to a ket: the Laplacian operator applied to a wave function: I ψ = ψ 2 ψ r = 2 x 2 ψ r + 2 y 2 ψ r + ψ r 2 z2 September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 19

20 Properties of operators Operators are said to be equal X = Y if X a = Y a for any arbitrary a in the ket-space. Addition operations: X + Y = Y + X Multiplication: X + Y + Z = X + Y + Z XY YX XYZ = X YZ = XY Z Linearity : A a ψ + b φ = aa ψ + ba φ for any a, b, ψ, φ September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 20

21 Remarks 1. The expectation value or mean value A of an operator A with respect to a state ψ is defined by ψ A ψ A = ψ ψ 2. The outer product: φ ψ ket () bra The quantity φ ψ is a linear operator. When it acts on a ket, φ ψ ψ = ψ ψ φ = c φ 3. Products of the type are forbidden. ψ A, A ψ September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 21

22 Relation between KET and BRA: Hermitian adjoint The products We define are not dual to each other. A ψ φ A A ψ φ A β X α = α X β (Dagger) The operator A is called the Hermitian adjoint of A. The Hermitian adjoint rules: constants: a = a Complex conjugate kets, bras: β = β, β = β operators: A = A ABC ψ = ψ C B A September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 22

23 Examples Which is correct? 1. α A = A α 2. φ ψ = φ ψ φ ψ = ψ φ 3. ca ψ = c ψ A = c ψ A = c ψ A September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 23

24 Hermitian operators When X = X, we call the operator X is a Hermitian operator. We have β X α = α X β β X α = β X α = α X β = α X β bra (c) ket bra (c) ket September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 24

25 Eigenkets and eigenvalues of an operator In general, A ψ = ψ There are particular kets known as eigenkets of an operator A a, a, a,, so that The set of numbers A a = a a, A a = a a, a, a, a, denoted by a is called a set of eigenvalues of operator A. A a = a a The physical state corresponding to an eigenket is called an eigenstate. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 25

26 Matrix representations An expansion of vectors in the Euclidean space in terms of the basis vectors r = a 1 i + a 2 j + a 3 k We can express any KET ψ of the Hilbert space in terms of orthonormal base kets: ψ = a 1 a 1 + a 2 a 2 + a 3 a 3 + a 4 a 4 + where the set of base kets a i satisfies (a) the orthonormality condition a i a j = δ ij (b) the completeness relation given by a n a n = I = n=1 the projection operator the identity operator δ ij = 0, i j 1, i = j September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 26

27 To represent the state vector ψ in the context of the basis a i, we expand the vector in terms of the base kets: ψ = I ψ = a n a n n=1 ψ = n=1 c n a n Hence, the ket can be represented by a column vector : ψ a 1 ψ a 2 ψ a 3 ψ a n ψ c n = c 1 c 2 c 3 c n September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 27

28 The bra ψ can be represented by a row vector: ψ ψ a 1 ψ a 2 ψ a n = a 1 ψ a 2 ψ a n ψ = c 1 c 2 c 3 c n In this representation, inner product is a complex number equal to the matrix product: ψ φ = c 1 c 2 c 3 c n b n = a n φ b 1 b 2 b 3 b n = c n b n n September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 28

29 Matrix representation of operators For each linear operator, we can write in the basis a i A = I A I = a n a n n=1 A a m a m m=1 A nm The matrix element are complex numbers: A nm = a n A a m = A nm a n a m n,m The matrix A is a square matrix, that represents operator A. A A = A 11 A 12 A 21 A 22 September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 29

30 Theorem: Hermitian operator The eigenvalues of a Hermitian operator A are real; the eigenkets of A form an orthogonal basis. Proof: First, recall that A a = a a For a Hermitian operator, we can write a A = a a where a, a, are eigenvalues of A. a A a = a a a a A a = a a a a a a a = 0 a = a when a = a the eigenvalues are REAL. a a when a a = 0; the eigenkets are orthogonal. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 30

31 Summary 2 Kets are represented by column vectors. Bras are represented by row vectors. Operators are represented by square matrices. Hermitian operators has real eigenvalues and orthogonal eigenkets. September 20, 2012 QM-1, L 2, U of Tsukuba, L. Bolotov 31