Quantum One

1. Quantum One

3. Continuously Indexed Basis Sets

4. In the last lecture, we began to describe a more general formulation of quantum mechanics, applicable to arbitrary quantum systems, which develops the basic postulates in a form that is designed to be representation independent. We began by stating the first postulate, which associated the dynamical state of a quantum system with a state vector|ψ〉that is an element of a complex linear vector spaceS. We then gave a definition of the term linear vector space, and saw that it defines a set of objects that we can multiply by scalars and add together, to obtain other elements of the set. That is, they obey a superposition principle. We then introduced a series of additional definitions, that included the idea of spanning sets, linearly independent sets, and basis sets, and we defined what we mean by the dimension of a linear vector space.

5. In the last lecture, we began to describe a more general formulation of quantum mechanics, applicable to arbitrary quantum systems, which develops the basic postulates in a form that is designed to be representation independent. We began by stating the first postulate, which associated the dynamical state of a quantum system with a state vector|ψ〉that is an element of a complex linear vector spaceS. We then gave a definition of the term linear vector space, and saw that it defines a set of objects that we can multiply by scalars and add together, to obtain other elements of the set. That is, they obey a superposition principle. We then introduced a series of additional definitions, that included the idea of spanning sets, linearly independent sets, and basis sets, and we defined what we mean by the dimension of a linear vector space.

6. In the last lecture, we began to describe a more general formulation of quantum mechanics, applicable to arbitrary quantum systems, which develops the basic postulates in a form that is designed to be representation independent. We began by stating the first postulate, which associated the dynamical state of a quantum system with a state vector|ψ〉that is an element of a complex linear vector spaceS. We then gave a definition of the term linear vector space, and saw that it defines a set of objects that we can multiply by scalars and add together, to obtain other elements of the set. That is, they obey a superposition principle. We then introduced a series of additional definitions, that included the idea of spanning sets, linearly independent sets, and basis sets, and we defined what we mean by the dimension of a linear vector space.

7. In the last lecture, we began to describe a more general formulation of quantum mechanics, applicable to arbitrary quantum systems, which develops the basic postulates in a form that is designed to be representation independent. We began by stating the first postulate, which associated the dynamical state of a quantum system with a state vector|ψ〉that is an element of a complex linear vector spaceS. We then gave a definition of the term linear vector space, and saw that it defines a set of objects that we can multiply by scalars and add together, to obtain other elements of the set. That is, they obey a superposition principle. We then introduced a series of additional definitions, that included the idea of spanning sets, linearly independent sets, and basis sets, and we defined what we mean by the dimension of a linear vector space.

8. In this lecture we continue our exploration of the mathematical properties of the state spaces of quantum mechanical systems. To this end we note that our previous definitions were expressed using a notation that was strictly applicable only to countable sets of states labeled by a discrete index. But we often encounter sets of vectors labeled by a continuous index . Examples from functional linear vector spaces include the plane waves and the delta functions. We therefore need to extend our definitions presented for discrete sets so that we can apply the same concepts to sets of vectors labeled by a continuous index.

9. In this lecture we continue our exploration of the mathematical properties of the state spaces of quantum mechanical systems. To this end we note that our previous definitions were expressed using a notation that was strictly applicable only to countable sets of states labeled by a discrete index. But we often encounter sets of vectors labeled by a continuous index . Examples from functional linear vector spaces include the plane waves and the delta functions. We therefore need to extend our definitions presented for discrete sets so that we can apply the same concepts to sets of vectors labeled by a continuous index.

10. In this lecture we continue our exploration of the mathematical properties of the state spaces of quantum mechanical systems. To this end we note that our previous definitions were expressed using a notation that was strictly applicable only to countable sets of states labeled by a discrete index. But we often encounter sets of vectors labeled by a continuous index . Examples from functional linear vector spaces include the plane waves and the delta functions. We therefore need to extend our definitions presented for discrete sets so that we can apply the same concepts to sets of vectors labeled by a continuous index.

11. In this lecture we continue our exploration of the mathematical properties of the state spaces of quantum mechanical systems. To this end we note that our previous definitions were expressed using a notation that was strictly applicable only to countable sets of states labeled by a discrete index. But we often encounter sets of vectors labeled by a continuous index . Examples from functional linear vector spaces include the plane waves and the delta functions. We therefore need to extend our definitions presented for discrete sets so that we can apply the same concepts to sets of vectors labeled by a continuous index.

12. In this lecture we continue our exploration of the mathematical properties of the state spaces of quantum mechanical systems. To this end we note that our previous definitions were expressed using a notation that was strictly applicable only to countable sets of states labeled by a discrete index. But we often encounter sets of vectors labeled by a continuous index . Examples from functional linear vector spaces include the plane waves and the delta functions. We therefore need to extend our definitions presented for discrete sets so that we can apply the same concepts to sets of vectors labeled by a continuous index.

13. Span- A continuously indexed set of vectors is said to span a vector space S if every vector |ψ〉 in Scan be written as a continuous linear combination of the elements of the set. In this expression the function gives the complex value of the expansion coefficient of multiplying the state of the spanning set. Linear Independence - A continuously indexed set of vectors is linearly independent if the only solution to the equation is for all α.

14. Span- A continuously indexed set of vectors is said to span a vector space S if every vector |ψ〉 in Scan be written as a continuous linear combination of the elements of the set. In this expression the function gives the complex value of the expansion coefficient of multiplying the state of the spanning set for the value of the continuous index. Linear Independence - A continuously indexed set of vectors is linearly independent if the only solution to the equation is for all α.

15. Span- A continuously indexed set of vectors is said to span a vector space S if every vector |ψ〉 in Scan be written as a continuous linear combination of the elements of the set. In this expression the function gives the complex value of the expansion coefficient of multiplying the state of the spanning set for the value of the continuous index. Linear Independence - A continuously indexed set of vectors is linearly independent if the only solution to the equation is for all α.

16. Span- A continuously indexed set of vectors is said to span a vector space S if every vector |ψ〉 in Scan be written as a continuous linear combination of the elements of the set. In this expression the function gives the complex value of the expansion coefficient of multiplying the state of the spanning set for the value of the continuous index. Linear Independence - A continuously indexed set of vectors is linearly independent if the only solution to the equation is for all vlues of the continuous index α.

17. Basis- A linearly independent set of continuously indexed vectors that spans S forms a basis for the space. We note in passing that any space that contains a continuously indexed basis, is necessarily infinite dimensional, since it must contain an infinite number of linearly independent vectors in any domain in which the index αtakes on continuous values.

18. Inner Products

19. Inner Products Towards a notion of length and direction

20. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

21. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

22. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

23. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

24. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

25. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

26. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

27. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

28. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

29. Another important property associated with the linear vector spaces of quantum mechanics is that they are inner product spaces. Definition: A linear vector space S is an inner product space if there exists an assignment to each pair of vectors and in , a scalar (an element of the field), denoted by the symbol , referred to as the inner product of and |ψ〉, obeying the following properties: 〈φ|φ〉 is real and non-negative, i.e., . Moreover, , if and only if is the null vector. Thus, the inner product distributes itself over vector addition. 〈φ|[λ|ψ〉]=λ〈φ|ψ〉 i.e., the inner produce distributes itself over scalar multiplication, and . Thus the order of the inner product is important for complex vector spaces.

30. In complex vector spaces, the inner product 〈φ|ψ〉 is linear in |ψ〉, but antilinearin. The first half of this comment follows from the observation that which follows from (2) and (3), while the second stems from the fact that ifthen which defines the condition of antilinearitywith respect to |φ〉.

31. In complex vector spaces, the inner product 〈φ|ψ〉 is linear in |ψ〉, but antilinearin. The first half of this comment follows from the observation that which follows from (2) and (3), while the second stems from the fact that ifthen which defines the condition of antilinearitywith respect to |φ〉.

32. In complex vector spaces, the inner product 〈φ|ψ〉 is linear in |ψ〉, but antilinearin. The first half of this comment follows from the observation that which follows from (2) and (3), while the second stems from the fact that ifthen which defines the condition of antilinearitywith respect to |φ〉.

33. In complex vector spaces, the inner product 〈φ|ψ〉 is linear in |ψ〉, but antilinearin. The first half of this comment follows from the observation that which follows from (2) and (3), while the second stems from the fact that ifthen which defines the condition of antilinearitywith respect to |φ〉.

34. In complex vector spaces, the inner product 〈φ|ψ〉 is linear in |ψ〉, but antilinearin. The first half of this comment follows from the observation that which follows from (2) and (3), while the second stems from the fact that ifthen which defines the condition of antilinearitywith respect to |φ〉.

35. In complex vector spaces, the inner product 〈φ|ψ〉 is linear in |ψ〉, but antilinearin. The first half of this comment follows from the observation that which follows from (2) and (3), while the second stems from the fact that ifthen which defines the condition of antilinearitywith respect to .

36. It is convenient to think of each vector as a column vector containing elements ψi, and to think of 〈φ| as a row vector whose elements are the complex conjugates of the components of the column vector representing . In this way the inner product can be viewed as the "dot product" This is, of course, the inner product commonly associated with CN. The complex conjugated row vectors associated with the symbols {〈φ|} thus form a vector space of their own, which is isomorphic (or dual or adjoint) to the original space S having elements { }. They are in 1-1 correspondence.

37. It is convenient to think of each vector as a column vector containing elements ψi, and to think of 〈φ| as a row vector whose elements are the complex conjugates of the components of the column vector representing . In this way the inner product can be viewed as the "dot product" This is, of course, the inner product commonly associated with CN. The complex conjugated row vectors associated with the symbols {〈φ|} thus form a vector space of their own, which is isomorphic (or dual or adjoint) to the original space S having elements { }. They are in 1-1 correspondence.

38. It is convenient to think of each vector as a column vector containing elements ψi, and to think of 〈φ| as a row vector whose elements are the complex conjugates of the components of the column vector representing . In this way the inner product can be viewed as the "dot product" This is, of course, the inner product commonly associated with CN. The complex conjugated row vectors associated with the symbols {〈φ|} thus form a vector space of their own, which is isomorphic (or dual or adjoint) to the original space S having elements { }. They are in 1-1 correspondence.

39. It is convenient to think of each vector as a column vector containing elements ψi, and to think of 〈φ| as a row vector whose elements are the complex conjugates of the components of the column vector representing . In this way the inner product can be viewed as the "dot product" This is, of course, the inner product commonly associated with CN. The complex conjugated row vectors associated with the symbols {〈φ|} thus form a vector space of their own, which is isomorphic (or dual or adjoint) to the original space S having elements { }. They are in 1-1 correspondence.

40. In the Dirac notation, an element of S is referred to as a ket, while an element 〈φ| of is referred to as a bra. The combination 〈φ|ψ〉 forms a "bracket", which in the Dirac formalism is always a number, i.e., an element of C. Examples: 1. In the space of displacement vectors in R³ the inner product is just the familiar "dot product". 2. As discussed above the inner product in CNis obtained by "dotting" a complex conjugated row vector into an unconjugated column vector.

41. In the Dirac notation, an element of S is referred to as a ket, while an element 〈φ| of is referred to as a bra. The combination 〈φ|ψ〉 forms a "bracket", which in the Dirac formalism is always a number, i.e., an element of C. Examples: 1. In the space of displacement vectors in R³ the inner product is just the familiar "dot product". 2. As discussed above the inner product in CNis obtained by "dotting" a complex conjugated row vector into an unconjugated column vector.

42. In the Dirac notation, an element of S is referred to as a ket, while an element 〈φ| of is referred to as a bra. The combination 〈φ|ψ〉 forms a "bracket", which in the Dirac formalism is always a number, i.e., an element of C. Examples: 1. In the space of displacement vectors in R³ the inner product is just the familiar "dot product". 2. As discussed above the inner product in CNis obtained by "dotting" a complex conjugated row vector into an unconjugated column vector.

43. In the Dirac notation, an element of S is referred to as a ket, while an element 〈φ| of is referred to as a bra. The combination 〈φ|ψ〉 forms a "bracket", which in the Dirac formalism is always a number, i.e., an element of C. Examples: 1. In the space of displacement vectors in R³ the inner product is just the familiar "dot product". 2. As discussed above the inner product in CNis obtained by "dotting" a complex conjugated row vector into an unconjugated column vector.

44. In the Dirac notation, an element of S is referred to as a ket, while an element 〈φ| of is referred to as a bra. The combination 〈φ|ψ〉 forms a "bracket", which in the Dirac formalism is always a number, i.e., an element of C. Examples: 1. In the space of displacement vectors in R³ the inner product is just the familiar "dot product". 2. As discussed above the inner product in CNis obtained by "dotting" a complex conjugated row vector into an unconjugated column vector.

45. 3. In functional spaces, the inner product involves the continuous analog of a summation over components, namely an integral. Thus, e.g., in the space of Fourier transformable function on R³ we "associate" with each function a vector . The inner product of two vectors then takes the form where the integral is over all space.

46. 3. In functional spaces, the inner product involves the continuous analog of a summation over components, namely an integral. Thus, e.g., in the space of Fourier transformable function on R³ we "associate" with each function a vector . The inner product of two vectors then takes the form where the integral is over all space.

47. The concept of an inner product allows us to make several new definitions: Norm- The positive real quantity is referred to as the norm, or the length of the vector A vector is said to be square-normalized, have unit norm, or be a unit vectorif Any vector having a finite norm can be square normalized. That is, if is not infinite, then the vector is a unit vector along the same direction in the space as |ψ〉

48. The concept of an inner product allows us to make several new definitions: Norm- The positive real quantity is referred to as the norm, or the length of the vector A vector is said to be square-normalized, have unit norm, or be a unit vectorif Any vector having a finite norm can be square normalized. That is, if is not infinite, then the vector is a unit vector along the same direction in the space as |ψ〉

49. The concept of an inner product allows us to make several new definitions: Norm- The positive real quantity is referred to as the norm, or the length of the vector A vector is said to be square-normalized, have unit norm, or be a unit vectorif Any vector having a finite norm can be square normalized. That is, if is not infinite, then the vector is a unit vector along the same direction in the space as |ψ〉

50. The concept of an inner product allows us to make several new definitions: Norm- The positive real quantity is referred to as the norm, or the length of the vector A vector is said to be square-normalized, have unit norm, or be a unit vectorif Any vector having a finite norm can be square normalized. That is, if is not infinite, then the vector is a unit vector along the same direction in the space as |ψ〉