Eigenvalues#
What you need to know
Linear vector spaces.
Dirac notation.
Functional spaces.
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 mathematical and physical meaning.
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 superimposed states.
Vectors, what are they?#
Let’s remind ourselves of some basics first. 3B1B has an excellent lecture series on linear algebra with stunning visual examples. I highly reccomend watching video 1 now and video 2 at the end of this chapter.
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.
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: $\(\mid a \rangle =(a_1,a_2,..) \,\,\,\,\,\)\( <br>Example: \)\(\mid a \rangle =(1, 2i)\)$ |
Ket: $\(\mid \psi\rangle=\psi\)\(<br> Example: \)\(\mid \psi \rangle=ix^2\)$ |
Bra:$\(\bra{a} = \begin{pmatrix}a_1 \\ a_2 \\ ...\\ \end{pmatrix}\)\( <br> \)\(\bra{a} =\begin{pmatrix} 1 \\ -2i \\ \end{pmatrix}\)$ |
Bra: $\(\langle \psi \mid=\psi^*\)\(<br> \)\(\langle \psi \mid = -ix^2\)$ |
$\(\langle a \mid b \rangle = \sum_i a_i b_i\)$ |
$\(\langle \phi \mid \psi \rangle = \int \phi(x)^* \psi(x) dx\)$ |
Example of Bra-Ket product for vectors in terms of components |
Example of Bra-Ket product for functions in terms of components |
Representation of vectors#
Notation for vectors can be different depending on the context. Below we list the different representation of the same vector.
- \[\vec{a}=2\vec{e_i}+3\vec{e_j}\]
- \[\mid a\rangle = 2\mid e_i\rangle+3 \mid e_j\rangle\]
- \[a=(2,3)\]
- \[\begin{split}a= \begin{pmatrix} 2 \\ 3 \\ \end{pmatrix}\end{split}\]
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.
However, a vector can be just an array of numbers, and the components of a vector can refer to quantities for which direction is less relevant e.g (age, height and weight) of a person, populations of some countries, stock prices over last ten years, etc.
- \[Person = (22, 1.75, 80)\]
- \[Population =(3.0, 40.0, 2.0, ...)\]
- \[Stock = (1.4, 3.6, 8.5, ...)\]
Vector operations#
What defines vectors is the operations on them. Let us take a simple 2D vector as an example:
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. If the projection is zero we say that the vectors are orthogonal. Example of the orthogonal vectors are unit vectors of cartesian coordinate system:
- \[\langle e_i \mid e_j \rangle =\delta_{ij}\]
- \[\begin{split}(1,0)\begin{pmatrix} 0\\ 1\\ \end{pmatrix}=1\cdot 0+0\cdot 1=0\end{split}\]
Where we denote both orthogonality and normalization with the convenient Kornecker symbol: \(\delta_{ij}=0\) when \(i\neq j\) and \(1\) when \(i=j\).
Norm of a vector \(\mid a\mid\) quantifies the length or magnitude of vector. Norm is defined via square root of dot product of vector with itself:
- \[\langle a \mid a\rangle= a_1^2+a_2^2\]
- \[\mid a \mid =\sqrt{a_1^2+a_2^2}\]
When the norm is \(\mid a \mid=1\), vector is called normalized. 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:
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.
Video “Linear combinations, span, and basis vectors”#
Now watch the second video from 3B1B on linear combinations and basis vectors.
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.
$\(\,\)$Sequence vectors |
Functions |
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Orthonormality of basis unit vectors x and y: |
Orthonormality of eigenfunctions of Hermitian operator: |
Linear superposition: |
Linear superposition: |
Coefficients are expressed via projections onto basis vectors: |
Coefficients are expressed via projections onto basis functions: |
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 are 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 \(\hat{H} \mid \psi_n \rangle=E_n \mid \psi_n \rangle\) are real:
Eigenfunctions are orthogonal:
Eigenfunctions form a complete basis set!
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, weather we want to express it in terms of the energy eigenfucntions or the position eigenfunctions, the key expressions are going to be the same!
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: \(f(x) = \sum_n c_n \Big(\frac{2}{L}\Big )^{1/2} sin \Big (\frac{n\pi x}{L} \Big )\)
What about coefficients \(c_n\)? They are what define the expansion. 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 ?
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 \(\hat{A}\).
Probabilites sum to one.
Averages are probability weighted sums of eigenvalues.#
Superposition is a legitimate stae in which 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.
Particle in a box example of a superoposition state#
Suppose a the particle in a box is in a state of supperopistion of 1-st and 5-th states of eigenfunctions of Hamiltonian:
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
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 trubles many physicsis, who were at the frontiers of development of quantum mechanics.
Act of an exeperimentation interferes with superposition state collapsing it to a particular eigenfunction with probability \(\mid c_n \mid^2\)
\[\mid \psi \rangle \rightarrow \mid \phi_n \rangle\]Orthogonality of eigenfunctions implies mutual exclusivity of system being in state 1 vs state 2
\[\langle \phi_1 \mid \phi_2 \rangle=0\]
Expectation values#
In most cases, we need to calculate expectation values for wavefunctions, which are not eigenfunctions of the given operator. It can be shown that for any given Hermitian operator and physically sensible boundary conditions, the eigenfunctions form a complete basis set. This means that any well-behaved function \(\psi\) can be written as a linear combination of the eigenfunctions \(\phi_i\) superposition state; the upper limit in the summation may be finite:
where \(c_i\) are constants specific to the given \(\psi\). Since the \(\phi_i\) are orthonormal (Eq. (\ref{eq9.43})) and \(\psi\)is normalized to one, we have:
The expectation value of \(\hat{A}\) is given (in terms of the eigenfunction basis; \(\hat{A}\) linear):
Above \(\left<\phi_i\left|\hat{A}\right|\phi_k\right>\) is often called a ``matrix element’’. Since \(\phi_i\)’s are eigenfunctions of \(\hat{A}\), we get:
Note that above \(\psi\) is not an eigenfunction of \(\hat{A}\). The expectation value is a weighted average of the eigenvalues.
The coefficients \(\left|c_i\right|^2\) give the probability for a measurement to give an outcome corresponding to \(a_i\). This is often taken as one of the postulates (``assumption’’) for quantum mechanics (Bohr’s probability interpretation). Note that the coefficients \(c_i\) may be complex but \(\left|c_i\right|^2\) is always real.
Given a wavefunction \(\psi\), it is possible to find out how much a certain eigenfunction \(\phi_i\) contributes to it (using orthogonality of the eigenfunctions):
Note that the discrete basis expansion does not work when the operator \(\hat{A}\) has a continuous set of eigenvalues (continuous spectrum).
The variance \(\sigma_A^2\) for operator \(\hat{A}\) is defined as:
The standard deviation is given by the square root of \(\sigma_A^2\).
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 \(\hat{H}\) (\(E_1 \ne E_2\)):
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 Since \(\psi\) is normalized and \(\phi_1\) and \(\phi_2\) are orthogonal, we have \(\left|c_1\right|^2 + \left|c_2\right|^2 = 1\). The probability of measuring \(E_1\) is \(\left|c_1\right|^2\) and \(E_2\) is \(\left|c_2\right|^2\). The average energy is given by:
Exercise: write the above equation without using the Dirac notation). 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):
Copenhagen interpretation#
“The Copenhagen interpretation is an expression of the meaning of quantum mechanics that was largely devised from 1925 to 1927 by Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, and remains one of the most commonly taught.(https://en.wikipedia.org/wiki/Copenhagen_interpretation#cite_note-Siddiqui2017-1)(https://en.wikipedia.org/wiki/Copenhagen_interpretation#cite_note-Wimmel1992-2)
According to the Copenhagen interpretation, physical systems generally do not have definite properties prior to being measured, and quantum mechanics can only predict the probability distribution of a given measurement’s possible results. The act of measurement affects the system, causing the set of probabilities to reduce to only one of the possible values immediately after the measurement. This feature is known as wave function collapse.”
Quantum superopsition of atom.#
Schordinger’s cat#
Schrödinger created a thought experiment to illustrate bizarre nature of quantum superpositions, in which a quantum system such as an atom or photon can exist as a combination of multiple states corresponding to different possible outcomes.
The thought Experiment puts cat in a box with a single radioactive atom whose state dictates weather it decays thereby breaking the poisonous chamber in the box that kills the cat or does not decay and cat stays alive. So Schrodinger argued kitty must be thought of simultaneously dead and alive until experiment is done and cat is found in one of the two states.