1.2. Basic Concepts in Quantum Mechanics

We will now review basic concepts in quantum mechanics, focusing on the application of linear algebra to quantum mechanics formalism.

1.2.1. Operators

In classical mechanics, an observable is described by a function. Instead, in quantum mechanics, an observable is represented by a linear operator that acts on a wavefunction. An operator is itself a function that, when applied to the state of a system, returns another function. A simplified view would describe it as an instruction to carry out some operation on a function to obtain the desired result - this may be anything ranging from a simple multiplication to differentiation, integration etc. (Strictly speaking, the linear operators used in quantum mechanics are special cases of a linear mapping \(V \rightarrow W\) where \(V = W\) - linear algebra plays a crucial role in quantum mechanics.)

In general, applying an operator \(\hat{\mathrm{A}}\) to some function \(f\) results in the function being changed to a new function, \(g\)

\[\begin{aligned} \hat{\mathrm{A}} f = g.\end{aligned}\]

One example is differentiation: considering \(\hat{\mathrm{A}} = d/dx\), and for example the function \(f=x^2\), the action of \(\hat{\mathrm{A}}\) on \(f\) is to transform it in a new function \(g=2x\)

\[\begin{aligned} \frac{d}{dx}\,\left(x^2\right) = 2x.\end{aligned}\]

1.2.1.1. Eigenfunctions and eigenvalues

A particular case is when the action of an operator \(\hat{\mathrm{A}}\) of a function \(f\) does not change the function, but results in the same function, multiplied by a scalar \(\lambda\):

\[\begin{aligned} \hat{\mathrm{A}} f = \lambda f.\end{aligned}\]

The multiplicative constant \(\lambda\) is the eigenvalue of the operator, and the function \(f\) is referred to as the eigenfunction of the operator \(\hat{\mathrm{A}}\). These terms are derived from the German word “eigen”, meaning “its own”, i.e. an eigenfunction of an operator is implied to belong to said operator. (We note en passant that the set of eigenvalues of an operator is called its spectrum.)

Together, all the eigenfunctions form a complete basis such that any general function \(g\) may be expressed in terms of a linear combination of the eigenfunctions:

\[\begin{split}\begin{aligned} \hat{\mathrm{A}} f_i &= \lambda_i f_i\\ g &= \sum_i c_if_i,\end{aligned}\end{split}\]

with \(c_i\) being the expansion coefficient of the function \(g\) in the complete basis of all the eigenfunctions \(f_i\)s.

The effect of an operator on an arbitrary function \(g\) can thus be expressed by decomposing \(g\) in terms of its basis functions:

\[\begin{aligned} \hat{\mathrm{A}}g = \hat{\mathrm{A}}\sum_i c_if_i = \sum_i c_i\hat{\mathrm{A}}f_i = \sum_i c_i \lambda_i f_i.\end{aligned}\]

Exercise 5

Find the eigenvalues and eigenvectors of the matrix \(\mathbf{A}\).

\[\begin{split} \mathbf{A} =\begin{aligned} \left(\begin{matrix} 2 & 1 \\ -1 & 4 \end{matrix}\right) \end{aligned} \end{split}\]

1.2.1.2. Commutator

For some operators such as simple multiplications, the order in which the operators are applied to the function does not matter. Consider this (trivial) example:

\[\begin{aligned} x\cdot y\cdot f(x) = y\cdot x\cdot f(x).\end{aligned}\]

Turning to derivatives, however, this is not the case anymore:

\[\begin{aligned} x\cdot \frac{\partial}{\partial x} f(x) \ne \frac{\partial}{\partial x} x\cdot f(x).\end{aligned}\]

It is trivial to see that it is indeed relevant in which order operators are applied to a function. We therefore define the commutator of two operators \(\hat{\mathrm{A}}\) and \(\hat{\mathrm{B}}\) as:

\[\begin{aligned} \left[\hat{\mathrm{A}},\hat{\mathrm{B}}\right] = \hat{\mathrm{A}}\hat{\mathrm{B}} -\hat{\mathrm{B}}\hat{\mathrm{A}} .\end{aligned}\]

Where the two operators are said to commute if their commutator is zero, i.e. \(\left[\hat{\mathrm{A}},\hat{\mathrm{B}}\right]=0\). If they commute, the order in which they are applied to a function does not matter - if they do not, i.e. if the commutator is nonzero, it does matter. Note that the commutator plays a crucial role in quantum mechanics, as it determines whether two observables can be simultaneously determined (cf. the Heisenberg uncertainty principle). Commuting operators share the same eigenfunctions (or, vice versa, operators sharing the same eigenfunctions will commute).

Exercise 6

Show that if the product \(\mathbf{C}=\mathbf{A}\mathbf{B}\) of two Hermitian matrices is also Hermitian, then \(\mathbf{A}\) and \(\mathbf{B}\) commute.

Exercise 7

Explain the connection between the Heisenberg uncertainty principle and the commutation relation.

1.2.2. Hermiticity and Matrix Representations

From here on, operators will be applied to some (wave)function \(\phi\), \(\psi\) rather than an arbitrary function \(f\). This is just a matter of notation and goes, of course, without loss of generality. (Note that we chose not to write \(\phi(x)\) or the like, because it is not relevant whether we use the position or the momentum wavefunction, cf. the bra-ket notation that will be discussed below.)

1.2.2.1. Hilbert Space and Bra-Ket Notation

Recalling that any arbitrary function can be expressed as a weighted superposition of eigenfunctions of their underlying operator, one needs to define a mathematical functional space in which these eigenfunctions reside. An N-dimensional euclidian space is restricted to real numbers, but the eigenfunctions we are interested in may as well be complex valued (Cartesian space is just an example of a 3-dimensional euclidian space).

The concept of euclidian space was generalised by Hilbert onto an infinite dimensional complex space of square integrable functions, the Hilbert space. (A Hilbert space is a Banach space; a vector space that has a well-defined metric such that the distance between two vectors as well as the magnitude of a vector can be computed.)

Each instantaneous state is described by a unit vector (meaning of norm 1) or a state vector in Hilbert space. Just like a position vector in cartesian space, the state vector is expressed in terms of the orthogonal basis vectors that are formed by all the eigenfunctions. (The chapter on Hermitian operators will justify this. One should also note that ‘all eigenfunctions’ refers to all eigenfunctions of an operator to which the Hilbert space under consideration is associated.) All these eigenfunctions taken together encode all the relevant information: Being a complete set (a basis), they are all that is needed to describe any arbitrary function \(g\).

A proper notation that accurately describes vectors with an infinite (but countable) number of elements - such as those in Hilbert space - was introduced by Paul Dirac. In bra-ket (or Dirac) notation, a vector in Hilbert space is denoted by a ket

\[\begin{aligned} \Ket{\Psi} = \left(\vdots\right), \end{aligned}\]

which is a column vector. In one-to-one correspondence to the space formed by the kets, there exists a dual space consisting of bra vectors:

\[\begin{aligned} \Bra{\Psi} = \left(\cdots\right), \end{aligned}\]

which is the adjoint of the ket (i.e. a row vector that is the complex conjugate of the ket). As both bra and ket are normalized vectors, their norm is 1. In Hilbert space, an inner product of two vectors \(\left<\Phi,\Psi\right>\) is always defined. For \(\Phi=\Psi\), the inner product defines the norm of the vectors: for two orthogonal basis vectors \(\psi_i,\psi_j\), we have that:

\[\begin{aligned} \left<\psi_i\middle|\psi_j\right> = \delta_{ij}, \end{aligned}\]

where \(\delta_{ij}\) denotes the Kronecker delta. Note that bras and kets themselves are not specified as being dependent on position or momentum, simply because the position and momentum wavefunctions are obtained by projecting their representation out of a ket in Hilbert space, as will be shown later. (However, time dependence of a state is often implicit.)

Applying bra-ket notation to the previously discussed description of any function in terms of eigenfunctions, any state of the system is given as a linear combination:

\[\begin{aligned} \Ket{\Psi} = \sum_j^{\infty} c_j\Ket{\psi_j}, \end{aligned}\]

where we introduced capital greek letters for the state of the system, and lower case letters for its basis (its eigenfunctions.) The expansion coefficients are simply the projections of the basis \(\psi_j\) onto the total state function \(\Psi\) (thinking of cartesian vectors, the projection of the x axis out of some vector yields the length of the vector on the x axis - the concept for a wavefunction expansion is the same).

\[\begin{aligned} c_j = \left<\psi_j\middle|\Psi\right> \end{aligned}\]

Because every state is normalised, just like the basis vectors it is built upon, any state \(\Psi\) must also be of the same norm. This requirement is conserved if and only if the normalisation \(\sum_j |c_j|^2 = 1\) is imposed.

Exercise 8

What is the meaning of a multiplication of a bra and a ket

\[ \begin{aligned} \left<a\middle|b\right>, \end{aligned} \]

and, conversely, an operator formed by a ket and a bra?

\[ \begin{aligned} \hat{\mathrm{O}} = \left|a\right>\left<b\right| \end{aligned} \]

Exercise 9

Given a basis \(\left\{\psi\right\}\) for which

\[ \begin{aligned} \left<\psi_i\middle|\psi_j\right> = \delta_{ij}, \end{aligned} \]

where \(\delta_{ij}\) is the Kronecker delta, for any state \(\Psi\)

\[ \begin{aligned} \Ket{\Psi} = \sum_j c_j\Ket{\psi_j}, \end{aligned} \]

the inner product is defined as

\[ \begin{aligned} \left<\Psi\middle|\Psi\right> = 1. \end{aligned} \]

Show that this holds only as long as

\[ \begin{aligned} \sum_j c_j^2 = 1, \end{aligned} \]

where the \(c_j\) are the aforementioned expansion coefficients.

Bonus Exercise 10

Prove that, given the above conditions, \(c_j=\left<\psi_j\middle|\Psi\right>\).

1.2.2.2. Linearity and Hermiticity

In order for an operator to be a valid quantum mechanical operator, i.e. physically observable, it must possess some properties:

• Linearity

\[\begin{aligned} \hat{\mathrm{A}}\left(\psi + \phi\right) = \hat{\mathrm{A}}\psi + \hat{\mathrm{A}}\phi. \end{aligned}\]

• Hermiticity (In the strict mathematical sense, this requirement implies linearity, but physicists often use the less specific definition of hermiticity as given below.) A Hermitian operator is defined as an operator that is equal to its conjugate transpose (Strictly speaking, this is true for an operator in a finite vector space \(V\); a Hermitian operator being a special case of a self-adjoint operator. Hermitian operators are normal operators, whence the spectral theorem holds.) $\(\begin{aligned} \hat{\mathrm{A}}^T = \overline{\hat{\mathrm{A}}}.\end{aligned}\)$ The eigenvalues of a Hermitian operator have some useful properties; they are real (which is a direct consequence of the definition of a Hermitian operator, as it shall be pointed out later again) and orthogonal to each other (this is why basis vectors in Hilbert space are orthogonal - and this is what allows for algebraic transformations in euclidian space to be easily extended onto the complex Hilbert space).

Exercise 11

Diagonalise the matrices \(\mathbf{A}\) and \(\mathbf{B}\). Specify which one is Hermitian.

\[\begin{split} \mathbf{A}=\begin{aligned} \left(\begin{matrix} 1 & 1-i \\ 1+i & 2 \end{matrix}\right) , \quad \mathbf{B} =\left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right) \end{aligned} \end{split}\]

Bonus Exercise 12

Prove that the eigenvalues of a Hermitian operator are real.

Exercise 13

Give the position and the momentum operators (consider only one dimension) in the position representation.

Exercise 14

Give the commutator of the position and linear momentum operators in the position representation (consider one dimension only).

Exercise 15

Is the electronic Hamiltonian \(\hat{H}_{el}\) a linear operator and why (not)?

1.2.2.3. Matrix Representation of Operators

The definition of a Hermitian operator being based on properties known from matrices implies a link between the two. Indeed, any operator can be written in a matrix form.

Consider another state \(\Phi\) that is defined analogously to \(\Psi\):

\[\begin{aligned} \Ket{\Psi} = \sum_j^{\infty} c_j\Ket{\psi_j} \quad ; \quad \Ket{\Phi} = \sum_k^{\infty} d_k\Ket{\psi_k}, \end{aligned}\]

and an operator such that

\[\begin{aligned} \Ket{\Phi} = \hat{\mathrm{A}}\Ket{\Psi}. \end{aligned}\]

(Since a Hermitian operator is a linear map \(V \rightarrow V\), the basis remains unchanged, but the expansion coefficients differ.) The matrix elements of the operator \(\hat{\mathrm{A}}\) in the basis \(\left\{\psi\right\}\) are now given as

\[\begin{aligned} A_{ij} = \Bra{\psi_i}\hat{\mathrm{A}}\Ket{\psi_j}, \end{aligned}\]

describing how the operator acts between the orthogonal basis vectors \(\psi_i\) and \(\psi_j\).

From the above, we deduce that

\[\begin{aligned} \sum_k^{\infty} d_k\Ket{\psi_k} = \hat{\mathrm{A}} \sum_j^{\infty} c_j\Ket{\psi_j}, \end{aligned}\]

and after multiplication by \(\sum_i^{\infty}\bra{\psi_i}\), exploiting orthogonality of the basis and applying the above definition of the operator’s matrix elements, it follows that:

\[\begin{aligned} d_i = \sum_j A_{ij}c_j, \end{aligned}\]

where the summation over an infinite but countable number of elements is implicitly assumed. This equation can be rewritten in matrix form:

\[\begin{split}\begin{aligned} \left(\begin{matrix} d_1 \\ d_2 \\ \vdots \\ d_i \end{matrix}\right) = \left(\begin{matrix} A_{11} & A_{12} & \cdots & A_{1j} \\ A_{21} & A_{22} & \cdots & A_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ A_{i1} & A_{i2} & \cdots & A_{ij} \end{matrix}\right) \left(\begin{matrix} c_1 \\ c_2 \\ \vdots \\ c_j \end{matrix}\right), \end{aligned}\end{split}\]

where \(c,d\) are vectors of expansion coefficients of \(\Psi,\Phi\) respectively. Operators for which \(A_{ij} = 0\) for \(i \neq j\) are, just like their underlying matrix, diagonal.

Exercise 16

Show that, if two operators \(\hat{\mathrm{A}}\), \(\hat{\mathrm{B}}\) commute and if \(\Ket{\psi}\) is an eigenvector of \(\hat{\mathrm{A}}\), \(\hat{\mathrm{B}}\Ket{\psi}\) is an eigenvector of \(\hat{\mathrm{A}}\), too, with the same eigenvalue.

Bonus: If \(\Ket{\psi}\) is part of a set of degenerate eigenvectors, show that the subspace spanned by the eigenvalues of \(\hat{\mathrm{A}}\) is invariant under the action of \(\hat{\mathrm{B}}\).

Exercise 17

Demonstrate that, if two hermitian operators \(\hat{\mathrm{A}}\), \(\hat{\mathrm{B}}\) commute and \({\Ket{\psi_1}}\), \({\Ket{\psi_2}}\) are eigenvectors of \(\hat{\mathrm{A}}\) associated to different eigenvalues, then the matrix element \(\Bra{\psi_1}\hat{\mathrm{B}}\Ket{\psi_2}\) vanishes.

1.2.3. The Wavefunction and its Properties

So far, any state \(\Psi\) was discussed as an abstract unit vector in Hilbert space that is expanded over a basis of eigenfunctions. The familiar picture of a wavefunction in position representation can be extracted from Hilbert space. The theory outlined so far will be applied to the energy eigenvalue problem of Schrödinger (the *time-independent Schrödinger equation).

1.2.3.1. The Wavefunction in Position Representation

The more abstract bra-ket notation and the notation of the wavefunction as being position or momentum dependent are often used interchangeably. Position and momentum wavefunctions are obtained by projecting the state function in Hilbert space onto position or momentum space using the respective operators \(\hat{\mathbf{r}}\) and \(\hat{\mathbf{p}}\). First, we note that the (non-square integrable) real space basis at a point \(\mathbf{r}_0\) is given by a delta function, which is the eigenfunction of the real space operator \(\hat{\mathbf{r}}_0\): $\(\begin{aligned} \xi_{\mathbf{r}_0}(\mathbf{r}) = \delta(\mathbf{r}-\mathbf{r}_0) \end{aligned}\)$

In bra-ket notation, we shall denote this basis function \(\Ket{\mathbf{r}_0}\). The corresponding projection operator \(\hat{P}\) is given by:

\[\begin{aligned} \hat{P}_{\mathbf{r}_0} = \Ket{\mathbf{r}_0}\Bra{\mathbf{r}_0}, \end{aligned}\]

with the closing relation:

\[\begin{aligned} \int \Ket{\mathbf{r}_0}\Bra{\mathbf{r}_0} \mathrm{d}\mathbf{r}_0 = \mathbb{1}, \end{aligned}\]

where \(\mathbb{1}\) is a unit matrix. Applying the projection operator onto the wavefunction, one has $\(\begin{aligned} \Ket{\psi} = \int \Ket{\mathbf{r}_0} \left<\mathbf{r}_0\middle|\Psi\right> \mathrm{d}\mathbf{r}. \end{aligned}\)$

The representation in the momentum basis can be developed analogously. The inner product \(\left<\mathbf{r}_0\middle|\Psi\right>\) can be interpreted as the coefficient of the expansion of \(\Ket{\Psi}\) on the basis \(\Ket{\mathbf{r}_0}\). After multiplication by a suitably chosen ket and making use of the orthogonality of the basis, one then readily finds:

\[\begin{split}\begin{aligned} \Psi\left(\mathbf{r},t\right) &= \left<\mathbf{r}\middle|\Psi\right>\\ \Psi\left(\mathbf{p},t\right) &= \left<\mathbf{p}\middle|\Psi\right>, \end{aligned}\end{split}\]

where the position and momentum vectors \(\mathbf{r}\) and \(\mathbf{p}\) are 3-dimensional. (Often, \(\mathbf{r}\) is just used for the length of a vector in position representation, and \(\mathbf{q}\) is used for coordinates. We will however resort to using \(\mathbf{r}\) throughout this script to denote a vector of (cartesian) coordinates.) Position and momentum wavefunction still contain all the information we need, and they conserve the properties of unit vectors in Hilbert space as discussed in the preceeding sections. Specifically, the \(\text{L}^2\) norm in real space is naturally given by:

\[\begin{aligned} \left<\Psi\middle|\Psi\right> = \int \Psi^\ast\left(\mathbf{r},t\right)\Psi\left(\mathbf{r},t\right)\mathrm{d}\mathbf{r}. \end{aligned}\]

Bonus Exercise 18

Show that the potential energy operator \(\hat{\mathrm{V}}(\mathbf{r})\) is multiplicative when applied to the real-space wavefunction.

Bonus Exercise 19

The link between position and momentum representation is given by a Fourier transform. Explain how this relates to the Heisenberg uncertainty principle.

1.2.3.2. Probabilities and Observables

Whereas a wavefunction contains all the neccessary information, it lacks any direct physical interpretation. Its square, however, fulfills the requirements of a probability distribution. Namely, for a many-electron wavefunction, where \(N\) electrons are associated to \(N\) coordinate vectors \(\left\{\mathbf{r}_i\right\}\):

\[\begin{aligned} P\left(\mathbf{r}_1,\mathrm{d}\mathbf{r}_1;\dots;\mathbf{r}_N,\mathrm{d}\mathbf{r}_N\right) = \left|\Psi\left(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N\right)\right|^2\mathrm{d}\mathbf{r} = \left|\Psi\left(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N\right)\right|^2\mathrm{d}\mathbf{r}_1\mathrm{d}\mathbf{r}_2\dots \mathbf{d}\mathbf{r}_N. \end{aligned}\]

The square of the wavefunction corresponds to the probability of finding all \(N\) electrons simultaneously in the infinitesimal volume elements \(\mathrm{d}\mathrm{r}_1,\mathrm{d}\mathrm{r}_2,\dots, \mathrm{d}\mathrm{r}_N\). The probability of finding electron 1 in \(\mathrm{d}\mathbf{r}_1\) and all the other electrons anywhere else is given by the N-1 dimensional integral:

\[\begin{aligned} P\left(\mathbf{r}_1,\mathrm{d}\mathbf{r}_1\right) = \mathrm{d}\mathbf{r}_1 \idotsint \Psi^\ast\left(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N\right)\Psi\left(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N\right) \mathrm{d}\mathbf{r}_2\dots\mathrm{d}\mathbf{r}_N \end{aligned}\]

Similarly, \(P\left(\mathbf{r}_1,\mathrm{d}\mathbf{r}_1;\mathbf{r}_2,\mathrm{d}\mathbf{r}_2\right)\) will be given by a N-2 dimensional integral.

All physically measurable quantities, the observables, are associated to a corresponding Hermitian operator \(\hat{\mathrm{O}}\) (it is common to denote operators that correspond to an observable by the letter \(O\)). Applying this operator to the state function of a system returns its eigenvalue if the state function is an eigenfunction of the operator; for a generic function \(\Psi\), each eigenvalue of the underlying basis \(\left\{\psi\right\}\) may be obtained, with a probability depending on the expansion coefficient \(c_i\) of the eigenfunction. The mean or expectation value of an observable \(O\) is then given by:

\[\begin{aligned} \left<\hat{\mathrm{O}}\right> = \frac{\Bra{\Psi}\hat{\mathrm{O}}\Ket{\Psi}}{\left<\Psi\middle|\Psi\right>} = \Bra{\Psi}\hat{\mathrm{O}}\Ket{\Psi} \end{aligned}\]

where the last equation can be retrived considering the normalisation condition.

From the properties of a Hermitian operator, we have that

\[\begin{aligned} \int \Phi^\ast\left(\mathbf{r}\right)\hat{\mathrm{O}}\Psi\left(\mathbf{r}\right)\mathrm{d}\mathbf{r} = \left( \int \Psi^\ast\left(\mathbf{r}\right)\hat{\mathrm{O}}\Phi\left(\mathbf{r}\right)\mathrm{d}\mathbf{r} \right)^\ast \end{aligned}\]

which proves that its eigenvalues are real, as stated earlier. This is a requirement for a physical observable and demonstrates why Hermitian operators are of great value in quantum mechanics.

1.2.3.3. Bosons and Fermions: Use of Determinants to Describe Many-Body Wavefunctions

Quantum mechanics imposes certain symmetry constraints on permutations of indistinguishable particles. For bosons, the wavefunction must remain unchanged upon exchange of particles. For fermions (such as electrons) instead, it has to change sign for an odd number of permutations. Formally,

\[\begin{aligned} \wp \Psi\left(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N\right) = \epsilon_p \Psi\left(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N\right),\end{aligned}\]

such that \(\epsilon_p = +1\) and \(\epsilon_p = -1\) for an even and an odd number of permutations respectively. This poses a problem for the construction of a many-particle (many-electron) wavefunction.
 
If one wishes to write a many-particle wavefunction as a simple product of one-particle wavefunctions

\[\begin{aligned} \Phi\left(\mathbf{r}_1,\dots,\mathbf{r}_N\right) = \prod_i^N \phi_i\left(\mathbf{r}_i\right), \end{aligned}\]

the resulting wavefunction will be even upon permutation of a single electron:

\[\begin{aligned} \Phi(\textbf{r}_1,\textbf{r}_2)=\phi_1(\textbf{r}_1) \phi_2(\textbf{r}_2) \neq - \phi_1(\textbf{r}_2) \phi_2(\textbf{r}_1) = -\Phi(\textbf{r}_2,\textbf{r}_1). \end{aligned}\]

A simple product of single-particle wavefunctions clearly violates the permutation requirement. A determinant of a matrix, however, has exactly the desired property:

\[\begin{split}\begin{aligned} \det A = \begin{vmatrix} a_{11} & \cdots & a_{1N} \\ \vdots & \ddots & \vdots \\ a_{N1} & \cdots & a_{NN} \end{vmatrix} = \sum_{i=1}^{N!} \left(-1\right)^\alpha a_{1p_1}a_{2p_2}\dots a_{Np_N}, \end{aligned}\end{split}\]

where \(\alpha\) is the number of permuted pairs \((p_i,p_j)\) with \(p_i > p_j\) and \(i < j\); the sum runs over all permutations \((p_1,p_2,\dots,p_N)\) of the set \((1,2,\dots,N)\). Upon exchange of two columns, the determinant changes sign (which is easily demonstrated using a simple 2x2 determinant). Exploiting this property, one defines a general N-electron wavefunction:

\[\begin{split}\begin{aligned} \Phi\left(1,2,\dots,N\right) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_{1} \left(\mathbf{r}_1\right) & \phi_{2} \left(\mathbf{r}_1\right) & \cdots & \phi_{N} \left(\mathbf{r}_1\right) \\ \phi_{1} \left(\mathbf{r}_2\right) & \phi_{2} \left(\mathbf{r}_2\right) & \cdots & \phi_{N} \left(\mathbf{r}_2\right)\\ \vdots & \vdots & \ddots & \vdots \\ \phi_{1} \left(\mathbf{r}_N\right) & \phi_{2} \left(\mathbf{r}_N\right) & \cdots & \phi_{N} \left(\mathbf{r}_N\right) \end{vmatrix} \end{aligned}\end{split}\]

where the \(\phi_i\) are one-electron wavefunctions and \(\frac{1}{\sqrt{N!}}\) is a normalisation factor. For a two-electron wavefunction, this reduces to

\[\begin{split}\begin{aligned} \Psi\left(\mathbf{r}_1,\mathbf{r}_2\right) = \frac{1}{\sqrt{2}} \begin{vmatrix} \psi_1\left(\mathbf{r}_1\right) & \psi_2\left(\mathbf{r}_1\right) \\ \psi_1\left(\mathbf{r}_2\right) & \psi_2\left(\mathbf{r}_2\right) \end{vmatrix} \end{aligned}\end{split}\]

and

\[\begin{split}\begin{aligned} \Psi\left(\mathbf{r}_2,\mathbf{r}_1\right) = \frac{1}{\sqrt{2}} \begin{vmatrix} \psi_2\left(\mathbf{r}_1\right) & \psi_1\left(\mathbf{r}_1\right) \\ \psi_2\left(\mathbf{r}_2\right) & \psi_1\left(\mathbf{r}_2\right) \end{vmatrix} = \Psi\left(\mathbf{r}_1,\mathbf{r}_2\right) \end{aligned}\end{split}\]

which is in agreement with the permutation requirement.

1.2.4. Time-Independent Schrödinger Equation

The preceeding theoretical considerations shall now be put to good use. Until now, it has not been stated which operator of interest spans the Hilbert space. The time-independent Schrödinger equation is an eigenvalue problem that applies the system’s Hamiltonian (the sum of kinetic and potential energy)

\[\begin{aligned} \hat{\mathrm{H}} = \hat{\mathrm{T}}+\hat{\mathrm{V}} \end{aligned}\]

onto a wavefunction, returning the expectation value of the energy, the energy eigenvalue:

\[\begin{aligned} \hat{\mathrm{H}}\Ket{\Psi} = E\Ket{\Psi}. \end{aligned}\]

The time evolution of the system at any point is given by the time-dependent Schrödinger equation:

\[\begin{aligned} \hat{\mathrm{H}}\Ket{\Psi} = i\hbar\frac{\partial}{\partial t}\Ket{\Psi} \end{aligned}\]

Note that the former is an eigenvalue problem, the latter is not.

One may rewrite the time-independent Schrödinger equation for an arbitrary state \(\Psi\) in matrix form.

\[\begin{split}\begin{aligned} \hat{\mathrm{H}}\Ket{\Psi} &= E\Ket{\Psi} \\ \sum_j c_j \hat{\mathrm{H}}\Ket{\psi_j} &= E \sum_j c_j\Ket{\psi_j} \end{aligned}\end{split}\]

All constants can be moved in front of the operator, and after multiplying by the bra \(\bra{\psi_i}\), one obtains

\[\begin{aligned} \sum_j c_j\Bra{\psi_i}\hat{\mathrm{H}}\Ket{\psi_j} = \sum_j H_{ij}c_j = Ec_i. \end{aligned}\]

Consequently, one recovers an infinite-dimensional matrix eigenvalue problem:

\[\begin{split}\begin{aligned} \left(\begin{matrix} H_{11} & H_{12} & \cdots & H_{1j} \\ H_{21} & H_{22} & \cdots & H_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ H_{i1} & H_{i2} & \cdots & H_{ij} \end{matrix}\right) \left(\begin{matrix} c_1 \\ c_2 \\ \vdots \\ c_j \end{matrix}\right) = E \left(\begin{matrix} c_1 \\ c_2 \\ \vdots \\ c_i \end{matrix}\right). \end{aligned}\end{split}\]

For a general matrix eigenvalue problem, there exist \(n\) solutions with eigenvalues \(E_k\). Generalising to the above problem results in:

\[\begin{aligned} \sum_i H_{ij}c_{jk} = c_{ik}E_k. \end{aligned}\]

This introduces the additional index \(k\), and the final eigenvalue problem that is to be solved is given by:

\[\begin{split}\begin{aligned} \left(\begin{matrix} H_{11} & H_{12} & \cdots & H_{1j} \\ H_{21} & H_{22} & \cdots & H_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ H_{i1} & H_{i2} & \cdots & H_{ij} \end{matrix}\right) \left(\begin{matrix} c_{11} & c_{12} & \cdots & c_{1k} \\ c_{21} & c_{22} & \cdots & c_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ c_{j1} & c_{j2} & \cdots & c_{jk} \end{matrix}\right) = \left(\begin{matrix} c_{11} & c_{12} & \cdots & c_{1k} \\ c_{21} & c_{22} & \cdots & c_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ c_{i1} & c_{i2} & \cdots & c_{ik} \end{matrix}\right) \left(\begin{matrix} E_{1} & 0 & \cdots & 0 \\ 0 & E_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & E_{k} \end{matrix}\right). \end{aligned}\end{split}\]

The eigenvalue problem of the time-independent Schrödinger equation can now be solved as the classical eigenvalue problem of a matrix, by resorting to simple linear algebra.

Exercise 20

In a system that consists of only two states (such as an electron spin in a magnetic field, where the electron spin can be in one of two orientations), the Hamiltonian has the following matrix elements: \(H_{11}=a, \ H_{22}=b, \ H_{12}=d, \ H_{21}=d\). How would you determine the energy levels \(E\) and the eigenstates \(\mathbf{\Psi}\) of the system? (You do not need to solve this problem explicitly, merely outlining the procedure is sufficient.)