Eigenvalues

Eigenvalues#

What you need to know

Learning the abstract mathematical formalism brings simplicity, unity and clairty to the relationships in quantum mechanisms. On the example of simple and familiar 2D-3D vectors we will illustrate the concepts of basis set and linear superpositions.
We will then show how Dirac notation can liberate one from coordinate representations and explicit intergrals which may obscure the underlying physics.
With an abstract formalism we are able to fully appreciate the strange nature of quantum states which exist in a superoposition of states! We will touch upon Schrödinger’s cat and the double slit experiments to illustrate the strange nature of quantum states.

Vectors, what are they?#

Let’s remind ourselves what are vectors. 3B1B has an excellent lecture series on linear algebra with stunning visual examples. I highly reccomend watching video 1 and 2.

Vectors in 2D/3D#

An example of a vector is an ordered collection of numbers, e.g:
- \(a=(-2,8) \,\,\,\) A 2D vector.
- \(b=(1.34,4.23,5.98) \,\,\,\) A 3D vector.
- \(c=(1,-2,4i,3+2i) \,\,\,\) A 4D vector with complex components.
- \(f=(1,2,3,4,5,6 ...,\infty)\,\,\,\) An infinite-dimensional vector with integers as components.
Notation for vectors can be different depending on the context. Below we list the different representation of the same vector.

\[\vec{a}=2\vec{e_i}+3\vec{e_j}\]

\[a=(2,3)\]

\[\mid a\rangle = 2\mid e_i\rangle+3 \mid e_j\rangle\]
These vectors can encode information about position of a particle or population of countries or temeprature in different parts of the forest, etc.
We examine the last notation closely that used Diract bra and ket notation for vectors

Dirac notation#

Here, anticipating their immense usefulenss, we introduce Dirac notation for vectors and functions. At this point let us just get used to this new and fancy looking notation.

Dirac notation for vectors	Dirac notation for functions
Ket \(\mid a \rangle =(a_1,a_2,..)\\ \) Bra \(\bra{a} = \begin{pmatrix}a_1 \\ a_2 \\ ...\\ \end{pmatrix}\)	Ket \(\mid \psi\rangle=\psi(x)\\\) \(\\\) Bra \(\langle \psi \mid=\psi(x)^*\)
Example \(\\ \mid a \rangle =(1, 2i)\) \(\\\bra{a} =\begin{pmatrix} 1 \\ -2i \\ \end{pmatrix}\)	Example \(\\ \mid \psi \rangle=e^{ix^2}\) \(\\ \langle \psi \mid = e^{-ix^2}\)
dot product \(\langle a \mid b \rangle = \sum_i a_i b_i\)	dot product \(\langle \phi \mid \psi \rangle = \int \phi(x)^* \psi(x) dx\)
Example \( \\ \langle a \mid a \rangle=(1)(1)+(2i)(-2i)=5\)	Example \(\\ \langle \psi \mid \psi \rangle = \int^L_0 (ix^2)(-ix^2) dx=\frac{L^5}{5}\)

Representation of vectors#

In classical physics vectors are attached to a coordinate system with unit vectors (\(\vec{e_i}\)) and are drawn with an arrow to emphasize that vector has a direction in addition to magnitude. Below is an example of unit vectors in cartesian space where each vector is aligned alogn x, y and z axes.

\[e_1 = (1, 0, 0)\,\,\, e_2=(0,1,0), e_3=(0,0,1)\]

In different coordinate system the same vector will have different components

Vector operations#

What defines vectors is the operations on them. Let us take a simple 2D vector as an example:

\[\mid a\rangle=a_1\mid e_1\rangle+a_2\mid e_2\rangle\]

1. Addition or subtraction with another vector \(\mid b\rangle=\mid e_1\rangle\pm\mid e_2\rangle\):

\[\begin{split} a+b=\begin{pmatrix} 2\\ 3\\ \end{pmatrix}+\begin{pmatrix} 1\\ 1\\ \end{pmatrix}=\begin{pmatrix} 3\\ 4\\ \end{pmatrix}\end{split}\]
\[\mid a\rangle \pm \mid b\rangle=(a_1\pm b_1)\mid e_1\rangle+(a_2\pm b_2)\mid e_2\rangle\]

2. Mulitiplication by a scalar \(\alpha=10\):

\[\begin{split}\alpha \cdot a=10\begin{pmatrix} 2\\ 3\\ \end{pmatrix}=\begin{pmatrix} 20\\ 30\\ \end{pmatrix}\end{split}\]
\[\alpha \mid a\rangle=\alpha a_1\mid e_1\rangle+ \alpha a_2\mid e_2\rangle\]

3. Dot product with another vector \(\mid b\rangle\):

\[\begin{split}a\cdot b=(2,3)\begin{pmatrix} 1 \\ 1 \\ \end{pmatrix}=2 \cdot 1+3\cdot 1=5\end{split}\]
\[\langle a \mid b\rangle=a_1b_1+a_2b_2\]

Projection, orthogonality and norm#

Dot product \(\langle a\mid b \rangle\) quantifies the projection of vector \(a\) on \(b\) and vice-versa. That is, how much \(a\) and \(b\) have in common with each other in terms of direction in space. I

\[\langle e_i \mid e_j \rangle =\delta_{ij}\]

Norm of a vector \(\mid a\mid\) Is project of the vector onto itself and quantifies the length of the vector. When the norm is \(\mid a \mid=1\), we say that the vector is normalized.

\[\langle a \mid a\rangle= a_1^2+a_2^2\]

\[\mid a \mid =\sqrt{a_1^2+a_2^2}\]

Orthogonality If the projection of vector \(a\) on \(b\) is zero we say that the vectors are orthogonal. Example of the orthogonal vectors are unit vectors of cartesian coordinate system.

\[\begin{split} (1,0)\begin{pmatrix} 0\\ 1\\ \end{pmatrix}=1\cdot 0+0\cdot 1=0 \end{split}\]

Orthonormal vectors are both normalized and orthogonal. We denote orthornamilty condition with the convenient Kornecker symbol: \(\delta_{ij}=0\) when \(i\neq j\) and \(1\) when \(i=j\).

To normalize a vector is to divide the vector by its norm. \(\mid E_1\rangle = (4,0,0,0)\) is not normalized since \(\langle E_1\mid E_1\rangle = 4\) hence we divide by norm and obtain a normalized vector \(\mid e_1\rangle=\frac{1}{4}\mid E_1\rangle=(1,0,0,0)\). And now \(\langle E_1 \mid E_1\rangle=1\).

Basis set and linear independence.#

1. Every \(N\)-dimensional vector can be uniquely represented as a linear combination of \(N\) orthogonal vectors. And vice-versa: if a vector can be represented by \(N\) orthogonal vectors, it means that the vector is \(N\)-dimensional. A set of vectors in terms of which an arbitrary \(N\)-dimensional vector is expressed is called a basis set.

\[\mid v\rangle = \sum^{i=N}_{i=1} \mid e_i\rangle\]
\[\begin{split}a= \begin{pmatrix} 2\\ 3\\ \end{pmatrix} = 2\begin{pmatrix} 1\\ 0\\ \end{pmatrix}+3 \begin{pmatrix} 0\\ 1\\ \end{pmatrix}\end{split}\]
\[\begin{split}a= \begin{pmatrix} -1\\ 5\\ 8\\ \end{pmatrix} = -1\begin{pmatrix} 1\\ 0\\ 0\\ \end{pmatrix}+5 \begin{pmatrix} 0\\ 1\\ 0\\ \end{pmatrix}+8 \begin{pmatrix} 0\\ 0\\ 1\\ \end{pmatrix}\end{split}\]

2. Orthogonal vectors are linearly independent. This means that no member of a set of vectors can be expressed in terms of the others. Linear independence is exprsessed mathematically by having coefficients of the linear combination of 3D (4D, ND, etc) vectors to zero \(\alpha_1=\alpha_2=\alpha_3=0\) as the only way to satify zero vector equality:

\[\alpha_1 \mid e_1\rangle +\alpha_1 \mid e_2\rangle+\alpha_3 \mid e_3\rangle=0\]

The converse, when one of the coefificent \(\alpha_i\)can be non-zero immeaditely implies linear depenence, because one can divide by that coeficient \(\alpha_i\) and express the unit vector \(\mid e_i\rangle\) in terms of the others.

Decomposition of functions into orthogonal components#

Writing a vector in terms of its orthogonal unit vectors is a powerful mathematical technique which permeates much of quantum mechanics. The role of finite dimensional vectors in QM play the infinite dimensional functions. In analogy with sequence vectors which can live in 2D, 3D or ND spaces, the inifinite dimensional space of functions in quantum mathematics is known as a Hilbert space, named after famous mathematician David Hilbert. We will not go too much in depth about functional spaces other than listing some powerful analogies with simple sequence vectors.

Vectors	Functions
Orthonormality \(\\ \langle x\mid y \rangle = \sum^{i=N}_{i=1} x_i y_i=\delta_{xy}\)	Orthonormality \(\\ \langle \phi_i \mid \phi_j \rangle = \int^{+\infty}_{-\infty} \phi_i(x) \phi_j(x)dx=\delta_{ij}\)
Linear superposition \(\\ \mid A \rangle = A_x \mid x\rangle+A_y\mid y\rangle\)	Linear superposition \(\\ \mid f\rangle = c_1 \mid\phi_1\rangle+c_2\mid\phi_2\rangle\)
Projections \(\\ \langle e_x\mid A\rangle=A_x \langle x\mid x \rangle +A_y \langle x\mid y \rangle=A_x \)	Projections \(\\ \langle \phi_1\mid \Psi\rangle=c_1 \langle \Psi \mid\phi_1 \rangle +c_2 \langle \Psi \mid\phi_2 \rangle=c_1\)

In the first column we decompose a vectors in terms of two orthogonal components \(A_i\) or projections of vector \(A\) along the orthonormal vectors \(x\) and \(y\). In the second column similiar decomposition where the dot product, due to infinite dimension, is given by an integral!

Eigenfunctions of Hermitian operators form complete basis set#

The three crucial consequences of Hermitian property of operators

\[\langle \phi \mid \hat{H} \mid \psi \rangle = \langle \psi \mid \hat{H}\mid \phi \rangle^*\]

Eigenvalues are real:

\[\hat{H} \mid \psi_n \rangle=E_n \mid \psi_n \rangle\]

\[E_n=E^*_n\]

Eigenfunctions are orthogonal (and can also be normalized)

\[\langle \psi_n \mid \psi_m\rangle=\delta_{nm}\]

Eigenfunctions form a complete basis set!

\[\mid f\rangle = \sum_i c_i \mid \psi_i \rangle\]

The last two properties imply that eigenfunctions of Hermitian opeartors play the same role for functions as the unit vectors for vectors. That is a function can be expressed in terms of the eigenfunctions of an opearators which can act on the function.

Wave function as a linear superoposition of eigenfunctions#

This is where we see the power and beautfy of Dirac notation. Reagardless of how the function \(f\) looks like, We can always express it in terms of the energy eigenfucntions or the position eigenfunctions or any other complete set of functions.

Express \(f(x)\) function in terms of eigenfunctions of \(\hat{H} \mid n\rangle=E_n \mid n \rangle\).
- In Dirac notation: \(f=\sum_n c_n \mid n\rangle\)
- In explicit notation: \(f(x) = \sum_n c_n \Big(\frac{2}{L}\Big )^{1/2} sin \Big (\frac{n\pi x}{L} \Big )\)
How do we find the coefficients \(c_n\)? Thanks to orthogonality of eigenfunctions any coeficient \(k\), just like component of a vector can be found by projecting our function (vectors) on eigenfunction \(k\) (unit basis vector \(k\)).
- In Dirac notation: \(c_k = \braket{k \mid f}\)
- In explicit notation: \(c_k = \Big(\frac{2}{L}\Big )^{1/2} \int sin \Big (\frac{k\pi x}{L} \Big )f(x) dx\)
Thus any wave function in quantum mechanics say \(f(x)=x^2\) on \([0,L]\) for particle in a 1D Box, can be expanded in terms of eigenfunctions of operators by plugging the function in above expression and finding the coefficeients which are what define the expansion. This is a mahematical fact. The next question is what is the physical signficance and meaning for the coefficeints and expansion.

Quantum states as linear superposition.#

Schrodinger equation as a linear differential equation admits as a general solution ithe linear superposition of eigenfunctions. This is a mathematical fact.
What is the physical meaning of solutions written as linear superpositions of eigenfunctions of some operator ?

\[\hat{A}\mid \phi_n \rangle = A_n \mid \phi_n \rangle\]

\[\mid \psi \rangle = \sum_n c_n \mid \phi_n \rangle \]

Absolue values of coeficients \(\mid c_n \mid^2\) are equal to probabilities \(p_n\) of finding system in a state \(n\) described by eigenvalue \(A_n\) and eigenfunction \(\mid \phi_n \rangle\) of the operator \(\hat{A}\).

\[p_n=\mid c_n \mid^2\]

Probabilites sum to one.

\[\sum_n \mid c_n \mid^2 =\sum_n p_n=1\]

Averages are probability weighted sums of eigenvalues.#

Superposition is a legitimate stae in which quantum objects can exist. For instance an atom can be in a superposition of ground and next excited states with 50% probabilities. Such a state is descibred by a normalized ket.

\[\mid \psi \rangle=c_1 \mid 1 \rangle+c_2 \mid 2\rangle\]

\[\begin{split}\langle \psi \mid \psi \rangle = \Big[c^*_1\langle 1\mid +c^*_2 \langle 2\mid \Big]\Big[c_1\mid 1\rangle + c_2 \mid 2\rangle\Big] =\\ = \mid c_1 \mid^2 \langle 1 \mid 1 \rangle+(c^*_1 c_2\langle 1 \mid 2 \rangle+c_1 c^*_2\langle 2 \mid 1 \rangle)+\mid c_2\mid^2 = c_1^2+c^2_2=p_1+p_2=1\end{split}\]

The meaning of expectation becomes more transparent as an average over all eigenvalues obtained in the experiment.

\[\begin{split}\langle E\rangle= \langle \psi \mid \hat{H}\mid \psi \rangle = \Big[c^*_1\langle 1\mid +c^*_2 \langle 2\mid \Big]\Big[c_1\hat{H}\mid 1\rangle + c_2 \hat{H}\mid 2\rangle\Big] =\Big[c^*_1\langle 1\mid +c^*_2 \langle 2\mid \Big]\Big[c_1E_1\mid 1\rangle + c_2 E_2\mid 2\rangle\Big] = \\ = c_1^2E_1+c^2_2 E_2=p_1E_1+p_2 E_2\end{split}\]

Example

Particle in a box is described as a supperopistion of 1-st and 5-th states.

Write down the wavefunction in terms of Hamiltonian operators eigenfunctions.
Compute the average energy

Solution

\[\psi(x)=\frac{1}{\sqrt{2}}\cdot \Big(\frac{2}{L} \Big )^{1/2}sin\frac{\pi x}{L}+\frac{1}{\sqrt{2}}\cdot \Big(\frac{2}{L} \Big )^{1/2}sin\frac{5\pi x}{L}\]

This means that when we measure energy we are going to obtain only two values \(E_1\) and \(E_5\) with equal probabilities \(p_1=p_2=(1/\sqrt{2})^2\).

The average of energy will be given by

\[\langle E \rangle =p_1 E_1+p_2 E_2 = \frac{1}{2}\frac{1^2 h^2}{8mL^2}+\frac{1}{2}\frac{5^2 h^2}{8mL^2}\]

Example

Consider a particle in a quantum state \(\psi\) that is a superposition of two eigenfunctions \(\phi_1\) and \(\phi_2\), with energy eigenvalues \(E_1\) and \(E_2\) of operator \(\hat{H}\) (\(E_1 \ne E_2\)):

\[\psi = c_1\phi_1 + c_2\phi_2\]

If one attempts to measure energy of such state, what will be the outcome?
What will be the average energy and the standard deviation in energy?

Solution

\[\left<\hat{H}\right> = \left<\psi\left|\hat{H}\right|\psi\right> = \left|c_1\right|^2\left<\phi_1\left|\hat{H}\right|\phi_1\right> + c_1^*c_2\left<\phi_1\left|\hat{H}\right|\phi_2\right> + c_2^*c_1\left<\phi_2\left|\hat{H}\right|\phi_1\right>\]

\[ + \left|c_2\right|^2\left<\phi_2\left|\hat{H}\right|\phi_2\right> = \left|c_1\right|^2E_1 + c_1^*c_2E_2\underbrace{\left<\phi_1\left|\phi_2\right.\right>}_{= 0} + c_2^*c_1E_1\underbrace{\left<\phi_2\left|\phi_1\right.\right>}_{= 0} + \left|c_2\right|^2E_2\]

\[= \left|c_1\right|^2E_1 + \left|c_2\right|^2E_2\]

The standard deviation is given by : \(\sigma_{\hat{H}} = \sqrt{\left<\hat{H}^2\right> - \left<\hat{H}\right>^2}\). We have already calculated \(\left<\hat{H}\right>\) above and need to calculate \(\left<\hat{H}^2\right>\) (use the eigenvalue equation and orthogonality):

\[\left<\hat{H}^2\right> = \left<\psi\left|\hat{H}^2\right|\psi\right> = \left<\psi\left|\hat{H}\right|E_1c_1\phi_1 + E_2c_2\phi_2\right> = \left<c_1\phi_1 + c_2\phi_2\left|E_1^2c_1\phi_1 + E_2^2c_2\phi_2\right.\right>\]

\[ = \left|c_1\right|^2E_1^2 + \left|c_2\right|^2E_2^2 \Rightarrow \sigma_{\hat{H}} = \sqrt{\left|c_1\right|^2E_1^2 + \left|c_2\right|^2E_2^2 - \left(\left|c_1\right|^2E_1 + \left|c_2\right|^2E_2\right)^2}\]

Quantum states as linear superposition of mutually exclusive states.#

It is important to emphasize that postulates of quantum mechanics that in an experimetn we always obtain one of the eigenvalues in other words the system described by a superoposition “collapses” to one of the eigenfunctions. The idea of a quantum system randomly collapsing into distinct and mutuallye esclusive states has trubled many physicsis, who were at the frontiers of development of quantum mechanics.