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1answer
67 views

Is the normalization of the wave-function preserved due to…?

Is the preservation of the inner product the same thing as the vector length of the wave-function staying constant with it's rotation through some R2 plane (ie it's evolution through time), that is ...
0
votes
1answer
43 views

Question on notation for the inner product of complex vectors [duplicate]

Regarding the wiki: https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form you can see that the wiki states that physics defines the inner product for complex vectors as: $$\langle \, \...
1
vote
2answers
93 views

Magnitude of the cross product of two bra-kets?

From the mathematical perspective, one can take the magnitude of a cross product: $$ |a\times b|=|a| |b| \sin{\theta}\cdot n, $$ where $\theta$ is the angle between a and b in the plane containing ...
0
votes
1answer
71 views

Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate]

In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by $$I = \sum_i |x_i\rangle\langle x_i|.$$ To me, ...
1
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0answers
75 views

Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]

I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions". Given a finite dimensional vector space with a basis $\{x_i\}$, I understand ...
0
votes
3answers
91 views

Why is $\langle c \cdot f|g\rangle=c^*\langle f|g\rangle$?

Why is $\langle c \cdot f|g\rangle=c^*\langle f|g\rangle$? $c$ is a complex number and $c^*$ is the conjugate. I think that $\langle c \cdot f|g\rangle=c\langle f|g\rangle$ because that's how scalar ...
0
votes
1answer
94 views

Prove Von Neumann entropy is invariant under coordinate transformation

https://en.wikipedia.org/wiki/Von_Neumann_entropy#Properties How to show that von Neumann entropy for $p_k$ with basis $|\psi_i\rangle$ is the same for $p_n$ with basis $|\phi_i\rangle$? That is, to ...
3
votes
1answer
259 views

What is the difference between a state vector and a basis vector in Quantum mechanics?

I searched about the difference between state vector and basis vector in Quantum mechanics but couldn't find any clear explanation. Can someone please give a simple and clear explanation of this?
2
votes
5answers
289 views

Why do we use vectors in quantum mechanics?

I've been trying to make my understanding of quantum mechanics more mathematically rigorous, but I'm struggling a bit with the lack of intuition behind the fact that we represent quantum states with ...
1
vote
0answers
54 views

The Field $\mathbb{F}$ of A Hilbert Space [duplicate]

Is it always necessary for the field of some arbitrary Hilbert space I define to describe a system be a field of complex numbers only? Is it possible to have a field of naturals, or reals? Since the ...
1
vote
1answer
35 views

Operators in infinite dimensions

In page 64 of Shankar's Principles of Quantum Mechanics, there are a few lines that leave me doubtful: It is worth remembering that $D_{xx'} = \delta'(x-x') $ is to be integrated over the second ...
1
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3answers
533 views

Hermitian operator in an orthonormal eigenbasis

In page 36 of Shankar's Principles of Quantum Mechanics is given a theorem: Theorem 10. To every Hermitian Operator $\Omega$, there exists (at least) a basis consisting of its orthonormal ...
0
votes
1answer
95 views

Hermitian operator in an eigenbasis

Does "Hermitian operator in an orthonormal eigenbasis" mathematically translate to, $$\sum_{i} \omega_{i} |i\rangle \langle i|$$ Where $\omega_{i}$ is an eigenvalue and $|i\rangle$ is a normalized ...
1
vote
0answers
70 views

Linear operators and the inner product

I pick the inner product involving the linear operator $\Omega$, $\langle i|\Omega|j\rangle$, from the $n\times n$ matrix $\Omega_{ij}$ as structured in page 21 of Principles of Quantum Mechanics by R....
-2
votes
2answers
320 views

Schwarz Inequality [closed]

The book—Principles of Quantum Mechanics by R.Shankar—proves the inequality in a way that leaves me unsettled. The proof is given in page 17. The author intends to compute $$\langle Z|Z \rangle =\...
0
votes
1answer
61 views

Inner product of vectors

While reading Shankar Ramamurti's book on Principles of Quantum Mechanics, p. 10, I came across the following lines under the concept of inner product of two vectors (in terms of orthonormal basis): ...
2
votes
1answer
223 views

How does a function represented in bra-ket notation become a vector of only coefficients?

I am learning quantum mechanics from the Miller Quantum Mechanics for Scientists and Engineers textbook. On page 97 it states that $$f(x)= \sum_{n}c_{n}\psi_{n}(x)$$ becomes $$|f(x)\rangle = \...
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3answers
113 views

Postulates of inner product

In quantum mechanics, two fundamental properties of inner products (J.J Sakurai) Chapter 1.2, are: $\langle \alpha|\beta\rangle = \langle \beta|\alpha\rangle^*$ $\langle \alpha|\alpha\rangle \ge 0$ ...
1
vote
1answer
93 views

Using integrals to expand a vector in continuous basis

I am new to quantum mechanics. I have been trying to understand why when we want to represent a function $$\psi(x)$$ as a ket in continuous basis |x> we us the integral: $$\vert \psi(x)\rangle =\int\...
1
vote
3answers
418 views

Very basic question about quantum field operators

For a matrix $A$, the notation $A^\dagger$ implies the transpose of the complex conjugate of $A$ i.e., $A^\dagger=(A^*)^T$. What does the symbol $\hat{\phi}^\dagger$ mean for a quantum operator ...
0
votes
2answers
105 views

Is a basis vector always unit-length in a wave function?

I'm currently studying wave functions and I came across an assertion, that $$\psi(x) = \left<x \middle| \psi \right>$$ is a projection of $\psi$ onto $x$. The vector projection being defined ...
0
votes
1answer
147 views

Proof that if expectation of an operator is zero for all vectors, then the operator itself must be zero

I was attending a Quantum Mechanics lecture when the instructor casually mentioned the following theorem: $\langle \alpha \rvert A \rvert \alpha \rangle = 0 ~\forall \alpha \implies A=0$, where $A$ ...
5
votes
5answers
775 views

Writing a vector as the sum of basis vectors

I'm currently making my way through quantum mechanics by Leonard Susskind, but have got stuck at this part; writing a vector as the sum of basis vectors. I get that for an $N$ dimensional space and a ...
0
votes
1answer
4k views

Proof of Schwarz Inequality using Bra-ket notation [closed]

I'm trying to prove Schwarz Inequality, where $$\mid\left\langle \alpha | \beta \right\rangle\mid^2 \leq \left\langle \alpha | \alpha\right\rangle \left\langle \beta| \beta\right\rangle $$ So I ...
4
votes
3answers
702 views

Rigorous mathematical definition of vector operator?

In standard quantum mechanics textbooks, the concept of operators is often introduced as linear maps that map a Hilbert space $H$ onto itself: $$ \hat{O}: H \rightarrow H \, . $$ However, directly ...
11
votes
4answers
830 views

Basis independence in Quantum Mechanics

The idea that the state of a system does not depend on the basis that we choose to represent it in, has always puzzled me. Physically I can imagine that the basis ought to just yield an equivalent ...
0
votes
0answers
358 views

What is the meaning of “closure is lost” for a set of kets (or any members of a vector space)?

This is the closure relation in Quantum Mechanics: $$\sum_i |i\rangle \langle i| = 1 $$ which I understand as "the sum of the projections onto the basis vectors leaves the projected vector unchanged"...
3
votes
3answers
818 views

What's the difference between classical and quantum vector superposition?

$(1)$Since quantum-mechanical states between two consecutive measurements are represented as superposition of orthonormal basis vectors in a vector space, at first glance it seems like it makes sense ...
0
votes
2answers
248 views

Dirac notation and column representation

$\renewcommand{ket}[1]{|#1\rangle}$ I am facing difficulty in understanding how the right hand side is coming in equation A below In $H$ of dimention 4, the vector $$ \sqrt{\frac{2}{3}} \ket{01} ...
4
votes
2answers
550 views

Why is the “complete metric space” property of Hilbert spaces needed in quantum mechanics?

I have been learning more about Hilbert spaces in an effort to better understand quantum mechanics. Most of the properties of Hilbert spaces seem useful (e.g. vector space, inner product, complex ...
9
votes
2answers
1k views

Infinite dimensional vector spaces vs. the dual space

I just happened across this over on Math Overflow. It references the following theorem from linear algebra: A vector space has the same dimension as its dual if and only if it is finite dimensional....
1
vote
2answers
167 views

Inner products with orthonormal bases

Probably a stupid question here - I think it's a case of me not having sufficient mathematical background to follow this through. In Leonard Susskind's Theoretical Minimum book, he represents the ket-...
1
vote
3answers
532 views

Why should multiplication of a ket vector by a complex number change only its “direction”?

Dirac argues on page 17 of his book, The Principles of Quantum Mechanics, that multiplication of a ket by a complex number shouldn't change the state this ket represents. But then concludes: Thus a ...
0
votes
1answer
417 views

Quantum states and state vectors

Does a state vector correspond to only one quantum states and the components in the state vector correspond to different states of this quantum state or is it that the components of the state vector ...
4
votes
2answers
1k views

Position Representation in Quantum Mechanics

How does the 3d position operator look like in position representation? I know that in 1d the position operator $\hat{x}$ is just multiplication by $x$.
1
vote
1answer
1k views

Bra space and adjoint vectors

If I'm not wrong, a bra, $ \langle \phi_n | $, can be thought as a linear functional that when applied to a ket vector, $| \phi_m \rangle$, returns a complex number; that is, the inner product it's a ...
3
votes
4answers
333 views

How to apply an algebraic operator expression to a ket found in Dirac's QM book?

I've been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac's book. In the fourth edition, 33rd page, starting from this:$$\xi|\xi'\rangle=\xi'|\xi'\rangle$$ (Where $\...