第八章 量子力学的矩阵形式与表象变换 2 量子力学教学课件
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Distributivity with respect to addition a(Ψ + Φ ) = a Ψ+a Φ, (a+b) Ψ = a Ψ+b Ψ.
Associativity with respect to multiplication of scalars a(b Ψ) = (ab) Ψ
exist a zero vector О, such that О+ Ψ= Ψ+ О = Ψ Existence of a symmetric or inverse vector: each vector Ψ must have a
symmetric vector – Ψ, Ψ+(- Ψ) = (- Ψ)+ Ψ
For each element Ψ there must exist a unitary scalar I and a zero scalar “0" such that I Ψ= ΨI= Ψ, 0 Ψ= Ψ0=0
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第4页
(Ψ, Ψ)>=0
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第5页
§1 量子态的不同表象，幺正变换
1.1 一矢量在两坐标系中的表示
x2
x’2
平面坐标系x1和x2的基矢e1和e2，长
A’2
度为1，彼此正交，即
A2
A
wenku.baidu.com
平面内任意矢量可表示为
e2 θ
O e’1
ee’21 θ A’1
if Ψ =a Ψ 1 +b Ψ 2: (Φ, a Ψ 1 +b Ψ 2) = a(Φ, Ψ 1 ) + b(Φ, Ψ 2) and antilinear with respect to the first factor if Φ =a Φ 1 +b Φ 2:
(aΦ 1 +b Φ 2, Ψ) = a (Φ 1, Ψ) + b (Φ 2, Ψ) The scalar product of a vector Ψ with itself is a positive real number:
第八章 量子力学的矩阵形式与表象变换 2 量子力学 教学课件
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun
概要
不同表象中，量子态、力学量的的表达方式不一样。 一组力学量完全集的共同本征态构成一组完备基底。 离散本征态，量子态——列矢；力学量——线性厄米矩阵。 量子力学中的各种问题皆可用线性代数的方法处理。 Dirac符号，脱离具体的表象，运算简洁，方便地推广到 连续表象。
If Ψ and Φ are vectors (elements) of a space, their sum, Ψ + Φ , is also a vector of the same space.
Commutativity: Ψ + Φ = Φ + Ψ Associativity: (Ψ + Φ)+ χ = Ψ + (Φ+ χ ) Existence of a zero or neutral vector: for each vector Ψ there must
The product of a scalar with a vector gives another vector. In general, if Ψ and Φ are two vectors of the space, any linear combination a Ψ +b Φ is also a vector of the space, a and b being scalars.
The scalar product of Ψ with Φ is equal to the complex conjugate of the
scalar product of Φ with Ψ: (Ψ, Φ) = (Φ, Ψ) *.
The scalar product of Φ with Ψ is linear with respect to the second factor
A1
x1
x’1
当A1, A2确定之后就确定了平面上一个矢量A。因此, (A1,A2)可以认 为就是矢量A在坐标系x1x2中的表示（列矢）。
The Hilbert Space
A Hilbert space H consists of a set of vectors (Ψ, Φ, χ……) and a set
of scalars (a, b, c……) which satisfy the following properties:
(a) H is a linear space
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第2页
The Hilbert Space and Wave Functions
(1) The Linear Vector Space
a set of vectors (Ψ, Φ, χ……) and a set of scalars (a, b, c……) a rule for vector addition and a rule for scalar multiplication (a) Addition rule
(b) H has a defined scalar product
The scalar product of an element Ψ with another element Φ is in general
a complex number, denoted by (Ψ, Φ), where (Ψ, Φ) = complex number.
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第3页
(b) Multiplication rule
The multiplication of vectors by scalars (scalars can be real or complex numbers) has these properties:
Associativity with respect to multiplication of scalars a(b Ψ) = (ab) Ψ
exist a zero vector О, such that О+ Ψ= Ψ+ О = Ψ Existence of a symmetric or inverse vector: each vector Ψ must have a
symmetric vector – Ψ, Ψ+(- Ψ) = (- Ψ)+ Ψ
For each element Ψ there must exist a unitary scalar I and a zero scalar “0" such that I Ψ= ΨI= Ψ, 0 Ψ= Ψ0=0
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第4页
(Ψ, Ψ)>=0
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第5页
§1 量子态的不同表象，幺正变换
1.1 一矢量在两坐标系中的表示
x2
x’2
平面坐标系x1和x2的基矢e1和e2，长
A’2
度为1，彼此正交，即
A2
A
wenku.baidu.com
平面内任意矢量可表示为
e2 θ
O e’1
ee’21 θ A’1
if Ψ =a Ψ 1 +b Ψ 2: (Φ, a Ψ 1 +b Ψ 2) = a(Φ, Ψ 1 ) + b(Φ, Ψ 2) and antilinear with respect to the first factor if Φ =a Φ 1 +b Φ 2:
(aΦ 1 +b Φ 2, Ψ) = a (Φ 1, Ψ) + b (Φ 2, Ψ) The scalar product of a vector Ψ with itself is a positive real number:
第八章 量子力学的矩阵形式与表象变换 2 量子力学 教学课件
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun
概要
不同表象中，量子态、力学量的的表达方式不一样。 一组力学量完全集的共同本征态构成一组完备基底。 离散本征态，量子态——列矢；力学量——线性厄米矩阵。 量子力学中的各种问题皆可用线性代数的方法处理。 Dirac符号，脱离具体的表象，运算简洁，方便地推广到 连续表象。
If Ψ and Φ are vectors (elements) of a space, their sum, Ψ + Φ , is also a vector of the same space.
Commutativity: Ψ + Φ = Φ + Ψ Associativity: (Ψ + Φ)+ χ = Ψ + (Φ+ χ ) Existence of a zero or neutral vector: for each vector Ψ there must
The product of a scalar with a vector gives another vector. In general, if Ψ and Φ are two vectors of the space, any linear combination a Ψ +b Φ is also a vector of the space, a and b being scalars.
The scalar product of Ψ with Φ is equal to the complex conjugate of the
scalar product of Φ with Ψ: (Ψ, Φ) = (Φ, Ψ) *.
The scalar product of Φ with Ψ is linear with respect to the second factor
A1
x1
x’1
当A1, A2确定之后就确定了平面上一个矢量A。因此, (A1,A2)可以认 为就是矢量A在坐标系x1x2中的表示（列矢）。
The Hilbert Space
A Hilbert space H consists of a set of vectors (Ψ, Φ, χ……) and a set
of scalars (a, b, c……) which satisfy the following properties:
(a) H is a linear space
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第2页
The Hilbert Space and Wave Functions
(1) The Linear Vector Space
a set of vectors (Ψ, Φ, χ……) and a set of scalars (a, b, c……) a rule for vector addition and a rule for scalar multiplication (a) Addition rule
(b) H has a defined scalar product
The scalar product of an element Ψ with another element Φ is in general
a complex number, denoted by (Ψ, Φ), where (Ψ, Φ) = complex number.
第8章
矩阵形式与表象变换@ Quantum Mechanics
Fang Jun 第3页
(b) Multiplication rule
The multiplication of vectors by scalars (scalars can be real or complex numbers) has these properties: