![Page 1: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/1.jpg)
Quantum One: Lecture 9
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Graham Schmidt Orthogonalization
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In the last lecture, we extended out definitions of spanning sets, linearly independent sets, and basis sets to allow an application of these concepts to continuously indexed sets of vectors.
We then introduced the idea of an inner product, which extends to complex vectors spaces the familiar dot product encountered in real vector spaces.
This allowed us to define the norm or length of a vector, to define unit vectors, and to introduce a limited notion of direction through the concept of orthogonality.
These notions of length, and orthogonality, allowed us to define orthonormal sets of vectors, with either discrete or continuous indices, and to end up with the idea of an orthonormal basis, i.e., an orthonormal set of vectors that span the space.
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In this lecture we begin by showing that it is always possible to construct an orthonormal basis set from any set of basis vectors of finite length.
The explicit algorithm for doing so, referred to as the Gram-Schmidt orthogonalization procedure, is presented below.
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Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
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Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
![Page 8: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/8.jpg)
Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
![Page 9: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/9.jpg)
Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
![Page 10: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/10.jpg)
Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
![Page 11: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/11.jpg)
Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
![Page 12: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/12.jpg)
Let be a set of linearly independent vectors of finite length.
1. Set
This produces a unit vector |φ₁ ⟩ pointing along the same direction as |χ₁ . ⟩
Now construct a second vector orthogonal to the first, by subtracting off that part of it which lies along the direction of the first vector:
2. Set
Note, that by construction
so |ψ₂⟩ and thus |φ₂ ⟩ are orthogonal to |φ₁ . ⟩
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We now proceed in this fashion, constructing each new vector orthogonal to each of those previously constructed. Thus,
3. Set
and, at the nth step
Set
so that
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We now proceed in this fashion, constructing each new vector orthogonal to each of those previously constructed. That is we
3. Set
and, at the nth step
Set
so that
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We now proceed in this fashion, constructing each new vector orthogonal to each of those previously constructed. Thus,
3. Set
and, at the nth step
Set
so that
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We now proceed in this fashion, constructing each new vector orthogonal to each of those previously constructed. Thus,
3. Set
and, at the nth step
Set
so that
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We now proceed in this fashion, constructing each new vector orthogonal to each of those previously constructed. Thus,
3. Set
and, at the nth step
Set
so that
![Page 18: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/18.jpg)
We now proceed in this fashion, constructing each new vector orthogonal to each of those previously constructed. Thus,
3. Set
and, at the nth step
Set
so that
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The only way this process could stop is if one of the resulting vectors turned out to be the null vector. A close inspection of the process reveals that this can't happen if the original set is linearly independent, as we have assumed.
Thus, in this way we can construct an orthonormal set of vectors equal in number to those of the original set.
It follows, that given any basis for the space we can construct an orthonormal basis with an equal number of vectors.
In the next lecture, we figure out why that’s a very good thing.
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The only way this process could stop is if one of the resulting vectors turned out to be the null vector. A close inspection of the process reveals that this can't happen if the original set is linearly independent, as we have assumed.
Thus, in this way we can construct an orthonormal set of vectors equal in number to those of the original set.
It follows, that given any basis for the space we can construct an orthonormal basis with an equal number of vectors.
In the next lecture, we figure out why that’s a very good thing.
![Page 21: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/21.jpg)
The only way this process could stop is if one of the resulting vectors turned out to be the null vector. A close inspection of the process reveals that this can't happen if the original set is linearly independent, as we have assumed.
Thus, in this way we can construct an orthonormal set of vectors equal in number to those of the original set.
It follows, that given any basis for the space we can construct an orthonormal basis with an equal number of vectors.
In the next lecture, we figure out why that’s a very good thing.
![Page 22: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/22.jpg)
The only way this process could stop is if one of the resulting vectors turned out to be the null vector. A close inspection of the process reveals that this can't happen if the original set is linearly independent, as we have assumed.
Thus, in this way we can construct an orthonormal set of vectors equal in number to those of the original set.
It follows, that given any basis for the space we can construct an orthonormal basis with an equal number of vectors.
We now explore how orthonormal bases make our lives easier.
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Expansion of a Vector on an Orthonormal Basis
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Discrete Bases - Let the set form an orthonormal basis (or ONB) for the
space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for a unique set of expansion coefficients .
Q: How do we determine what these expansion coefficients are?
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Discrete Bases - Let the set form an orthonormal basis (or ONB) for the
space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for a unique set of expansion coefficients .
Q: How do we determine what these expansion coefficients are?
![Page 27: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/27.jpg)
Discrete Bases - Let the set form an orthonormal basis (or ONB) for the
space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for a unique set of expansion coefficients .
Q: How do we determine what these expansion coefficients are?
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Discrete Bases - Let the set form an orthonormal basis (or ONB) for the
space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for a unique set of expansion coefficients .
Question: How do we determine what these expansion coefficients are?
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
expansion coefficient = inner product with basis vector
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
expansion coefficient = inner product with basis vector
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
expansion coefficient = inner product with basis vector
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
expansion coefficient = inner product with basis vector
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
expansion coefficient = inner product with basis vector
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
We can then write the expansion as
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Consider the inner product of the vector |χ with an arbitrary element of this ⟩basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
We can then write the expansion as
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Extension to Continuous Bases - Let the set form an orthonormal basis (or ONB) for the space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for some unique expansion function .
Q: How do we determine what this expansion function is?
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Extension to Continuous Bases - Let the set form an orthonormal basis (or ONB) for the space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for some unique expansion function .
Q: How do we determine what this expansion function is?
![Page 38: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/38.jpg)
Extension to Continuous Bases - Let the set form an orthonormal basis (or ONB) for the space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for some unique expansion function .
Q: How do we determine what this expansion function is?
![Page 39: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/39.jpg)
Extension to Continuous Bases - Let the set form an orthonormal basis (or ONB) for the space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for some unique expansion function .
Q: How do we determine what this expansion function is?
![Page 40: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/40.jpg)
Extension to Continuous Bases - Let the set form an orthonormal basis (or ONB) for the space S, so that
and let |χ be an arbitrary element of the space. ⟩
By assumption there exists an expansion of the form
for some unique expansion function .
Question: How do we determine what this expansion function is?
![Page 41: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/41.jpg)
Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus, expansion coefficient = inner product with basis vector
![Page 42: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/42.jpg)
Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus, expansion coefficient = inner product with basis vector
![Page 43: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/43.jpg)
Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus, expansion coefficient = inner product with basis vector
![Page 44: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/44.jpg)
Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus, expansion coefficient = inner product with basis vector
![Page 45: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/45.jpg)
Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus, expansion coefficient = inner product with basis vector
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Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus, expansion coefficient = inner product with basis vector
![Page 47: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/47.jpg)
Consider the inner product of the vector |χ with an arbitrary element of ⟩this basis:
This shows that the expansion coefficient is just the inner product of the vector we are expanding with the unit vector along that direction in Hilbert space.
Thus,
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So: where
We will refer to the function χ(α) as the "wave function" representing |χ in the α ⟩representation.
Note: This expansion can also be written in the suggestive the form
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So: where
We will refer to the function χ(α) as the "wave function" representing |χ in the α ⟩representation.
Note: This expansion can also be written in the suggestive the form
![Page 50: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/50.jpg)
So: where
We will refer to the function χ(α) as the "wave function" representing |χ in the α ⟩representation.
Note: This expansion can also be written in the suggestive form
![Page 51: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/51.jpg)
So: where
We will refer to the function χ(α) as the "wave function" representing |χ in the α ⟩representation.
Note: This expansion can also be written in the suggestive form
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A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanded in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
![Page 53: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/53.jpg)
A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanded in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
![Page 54: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/54.jpg)
A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanding in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
![Page 55: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/55.jpg)
A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanding in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
![Page 56: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/56.jpg)
A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanding in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
![Page 57: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/57.jpg)
A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanded in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
![Page 58: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/58.jpg)
A Notational Simplification:
It is clear that when we talk about ONB's, such as } or , the ⟩important information which distinguishes the different basis vectors from one another is the label or index: i or j in the discrete case, α or α in the continuous ′case.
The symbols φ just sort of come along for the ride, a historical vestige of when we were expanding in sets of functions.
From this point on we will acknowledge this by using an abbreviated notation:
and
In this way the expansions of an arbitrary ket can be written
and
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Calculation of Inner Products Using an Orthonormal Basis
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The Emergence of Numerical Representations
![Page 63: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/63.jpg)
Discrete Bases - Let the set form an orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis set
Suppose we know these expansion coefficients, and we want to know the inner product of these two vectors.
![Page 64: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/64.jpg)
Discrete Bases - Let the set form an orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis set
Suppose we know these expansion coefficients, and we want to know the inner product of these two vectors.
![Page 65: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/65.jpg)
Discrete Bases - Let the set form an orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis set
Suppose we know these expansion coefficients, and we want to know the inner product of these two vectors.
![Page 66: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/66.jpg)
Discrete Bases - Let the set form an orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis set
Suppose we know these expansion coefficients, and we want to know the inner product of these two vectors.
![Page 67: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/67.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in in CN
![Page 68: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/68.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in in CN
![Page 69: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/69.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in in CN
![Page 70: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/70.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in in CN
![Page 71: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/71.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in in CN
![Page 72: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/72.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in CN
![Page 73: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/73.jpg)
Thus we have an important result:
Any discrete orthonormal basis {|i } ⟩ generates a column-vector/ row-vector representation, i.e., it gives us a natural way of associating each ket |ψ with a ⟩ complex-valued column vector with components , and each bra ψ| in S⟨ ∗ with a complex-valued row vector with components in terms of which the inner product of two states can be written
It is important to note that in our formulation, the quantum state vector |ψ is not ⟩a column vector, but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 74: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/74.jpg)
Thus we have an important result:
Any discrete orthonormal basis {|i } ⟩ generates a column-vector/ row-vector representation, i.e., it gives us a natural way of associating each ket |ψ with a ⟩ complex-valued column vector with components , and each bra ψ| in S⟨ ∗ with a complex-valued row vector with components in terms of which the inner product of two states can be written
It is important to note that in our formulation, the quantum state vector |ψ is not ⟩a column vector, but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 75: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/75.jpg)
Thus we have an important result:
Any discrete orthonormal basis {|i } ⟩ generates a column-vector/ row-vector representation, i.e., it gives us a natural way of associating each ket |ψ with a ⟩ complex-valued column vector with components , and each bra ψ| in S⟨ ∗ with a complex-valued row vector with components in terms of which the inner product of two states can be written
It is important to note that in our formulation, the quantum state vector |ψ is not ⟩a column vector, but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 76: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/76.jpg)
Continuous Bases - Let the set form a continuous orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis
Suppose we know these expansion functions, and we want to know the inner product of these two vectors.
![Page 77: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/77.jpg)
Continuous Bases - Let the set form a continuous orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis
Suppose we know these expansion functions, and we want to know the inner product of these two vectors.
![Page 78: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/78.jpg)
Continuous Bases - Let the set form a continuous orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis
Suppose we know these expansion functions, and we want to know the inner product of these two vectors.
![Page 79: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/79.jpg)
Continuous Bases - Let the set form a continuous orthonormal basis for S, so that
and let |χ⟩ and |ψ ⟩ be arbitrary states of S.
These states can be expanded in this basis
Suppose we know these expansion functions, and we want to know the inner product of these two vectors.
![Page 80: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/80.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in functional linear vector spaces
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Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in functional linear vector spaces
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Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in functional linear vector spaces
![Page 83: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/83.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in functional linear vector spaces
![Page 84: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/84.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in functional linear vector spaces
![Page 85: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/85.jpg)
Well, we can express the desired inner product in the form
But
So we can write this inner product in the form
But this is exactly of the form of the inner product in functional linear vector spaces
![Page 86: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/86.jpg)
Thus we have an important result:
Any continuous orthonormal basis {|α } induces a ⟩ wave function representation for the space, i.e., it gives us a natural way of associating each ket |ψ in S with a complex-valued wave function ψ(α) and each⟩bra ψ| in S⟨ ∗ with a complex-valued wave function ψ∗ (α). We refer to ψ(α) as the wave function for the state |ψ in the α-representation. In this representation the ⟩inner product of two states can be computed as
It is important to note that in our formulation, the quantum state vector |ψ is ⟩ not a wave function but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 87: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/87.jpg)
Thus we have an important result:
Any continuous orthonormal basis {|α } induces a ⟩ wave function representation for the space, i.e., it gives us a natural way of associating each ket |ψ in S with a complex-valued wave function ψ(α) and each⟩bra ψ| in S⟨ ∗ with a complex-valued wave function ψ∗ (α). We refer to ψ(α) as the wave function for the state |ψ in the α-representation. In this representation the ⟩inner product of two states can be computed as
It is important to note that in our formulation, the quantum state vector |ψ is ⟩ not a wave function but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 88: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/88.jpg)
Thus we have an important result:
Any continuous orthonormal basis {|α } induces a ⟩ wave function representation for the space, i.e., it gives us a natural way of associating each ket |ψ in S with a complex-valued wave function ψ(α) and each⟩bra ψ| in S⟨ ∗ with a complex-valued wave function ψ∗ (α). We refer to ψ(α) as the wave function for the state |ψ in the α-representation. In this representation the ⟩inner product of two states can be computed as
It is important to note that in our formulation, the quantum state vector |ψ is ⟩ not a wave function but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 89: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/89.jpg)
Thus we have an important result:
Any continuous orthonormal basis {|α } induces a ⟩ wave function representation for the space, i.e., it gives us a natural way of associating each ket |ψ in S with a complex-valued wave function ψ(α) and each⟩bra ψ| in S⟨ ∗ with a complex-valued wave function ψ∗ (α). We refer to ψ(α) as the wave function for the state |ψ in the α-representation. In this representation the ⟩inner product of two states can be computed as
It is important to note that in our formulation, the quantum state vector |ψ is ⟩ not a wave function but it may in this way be represented by or associated with one.
In fact, this may be done in an infinite number ways.
![Page 90: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/90.jpg)
In the next lecture we attempt to make some of these admittedly abstract ideas more concrete, by applying our general formulation of quantum mechanics to the quantum state space of a single quantum mechanical particle.
In so doing, we will see how, in a natural and physically meaningful way the “Schrödinger” representation, which associates the quantum state with a wave function , emerges from the formal mathematical structure developed thus far.
![Page 91: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/91.jpg)
In the next lecture we attempt to make some of these admittedly abstract ideas more concrete, by applying our general formulation of quantum mechanics to the quantum state space of a single quantum mechanical particle.
In so doing, we will see how, in a natural and physically meaningful way the “Schrödinger” representation, which associates the quantum state with a wave function , emerges from the formal mathematical structure developed thus far.
![Page 92: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/92.jpg)
In the next lecture we attempt to make some of these admittedly abstract ideas more concrete, by applying our general formulation of quantum mechanics to the quantum state space of a single quantum mechanical particle.
In so doing, we will see how, in a natural and physically meaningful way the “Schrödinger” representation, which associates the quantum state with a wave function , emerges from the formal mathematical structure developed thus far.
![Page 93: Quantum One: Lecture 9. Graham Schmidt Orthogonalization](https://reader037.vdocuments.mx/reader037/viewer/2022110319/56649c745503460f949265cf/html5/thumbnails/93.jpg)
In the next lecture we attempt to make some of these admittedly abstract ideas more concrete, by applying our general formulation of quantum mechanics to the quantum state space of a single quantum mechanical particle.
In so doing, we will see how, in a natural and physically meaningful way the “Schrödinger” representation, which associates the quantum state with a wave function , emerges from the formal mathematical structure developed thus far.