Questions tagged [quantum-mechanics]

When to Use The Tag

Use the tag quantum-mechanics when asking questions about small quantum mechanical systems, such as a single hydrogen atom, or general aspects of quantum mechanics, e.g., the uncertainty principle, the wave-particle duality, or simple scattering. Use quantum-field-theory when asking questions about many-body quantum mechanical systems. In most cases, there is no need to tag a question as both quantum-mechanics and quantum-field-theory.

Introduction

Single-body quantum mechanics is usually done in a Hilbert space $\mathcal{H}$ of states, where the Bra-Ket notation is used. $|\Psi\rangle$ refers to an element of $\mathcal{H}$ and $\langle \Psi |$ to an element of the dual, $\mathcal{H}^\star$. The state of a physical system corresponds to a ray in $\mathcal H$, that is, to the equivalence class of all vectors $|\Psi\rangle$ that differ by a phase. In this sense, the physics of a quantum mechanical system is realized on a Projective Hilbert space. Two vectors related by $|\Psi'\rangle=\mathrm e^{i\theta}|\Psi\rangle$ correspond to the same physical state.

The projection of a vector $|\Psi\rangle$ onto a particular basis gives a wave function in this basis, cf. wavefunction. For example, the wave function associated to the state $|\Psi\rangle$ in the basis $|\vec r\rangle$ is $$\langle \vec r | \Psi(t) \rangle = \Psi(\vec r,t).$$

In standard treatments of Quantum Mechanics, one often introduces the notion of a generalized vector, an example being $\langle \vec r|$. This vector has no representation in terms of an actual function, but in terms of a generalized function instead of a Dirac delta. Such an object is not normalizable, and it is more properly understood as an element of a rigged Hilbert space.

Operators, such as $\hat x$ and $\hat p$, are then defined to act on these states. Use the tag operators whenever your question focuses on this aspect of Quantum Mechanics.

The most prominent example of an operator is the Hamiltonian $\hat H$, which by definition determines the time evolution of the system. This operator has the possible energies of the system as eigenvalues. The eigenvalue equation of the Hamilton operator $\hat H$ is the so-called time-independent schrodinger-equation (sometimes referred to as TISE):

$$ \hat H |\Psi\rangle = E |\Psi \rangle.$$

While the Hilbert space $\mathcal H$ is, in general, a complex vector space (so that wave-function $\Psi(\vec r,t)$ can be complex), operators that correspond to physical observables ($\hat x$, $\hat p$, $\hat H$) must have real eigenvalues and real expectation values, implying that these operators are Hermitian. In other words, for any pair of states $i,j$ the matrix elements of $\hat A$ satisfy $$ \langle i|\hat A^{\dagger}|j\rangle =\langle j|\hat A|i\rangle^*=\langle i|\hat A|j\rangle$$ where the first equality holds by definition of $\dagger$, and the second one by definition of Hermitian.

The act of applying a projection operator onto the eigenspace of a hermitian operator, such as $\hat x$, on a state $|\Psi\rangle$ is taken to be equal to a measurement on the system described by $|\Psi\rangle$. The eigenvalue of the corresponding eigenspace is then the ‘measured’ value (in this case, the particle's position). Most of these operators relate to classical mechanical operators by the correspondence principle.

Equations of Motion

In the beginning, operators are usually taken to be constant, whereas the states $|\Psi\rangle$ evolve in time. This is known as the Schrödinger picture, and its time evolution is governed by the schrodinger-equation:

$$ i \hbar \frac{\mathrm d}{\mathrm d t} |\Psi(t)\rangle = \hat H | \Psi(t) \rangle .$$

An equivalent view is the Heisenberg picture, where states $|\Psi\rangle$ are assumed to be constant, and operators evolve in time according to

$$\frac{\mathrm{d}}{\mathrm{d}t} \hat A(t) = \frac{i}{\hbar} [ \hat H , \hat A(t) ] + \frac{\partial}{\partial t} A(t) .$$ where $[\cdot,\cdot]$ stands for commutator.

A third alternative is to consider the so-called Interaction picture (sometimes, Dirac picture). This picture is the most convenient one to formulate perturbation-theory, which deals with the task of systematically approximating observables to any desired degree of precision.

Path integral

Another equivalent formulation of Quantum Mechanics is the so-called Path Integral Formulation (or Feynman Formulation), which is as described in path-integral.

In the Path Integral formulation a functional called the phase is associated with each path $x=x(t)$:

$$\phi[x] = A \mathrm e^\frac{iS[x]}{\hbar} $$ where $S[x]$ is the action functional associated with a classical system.

The Kernel or the Matrix Element is the path integral of this phase.

$$K(x_1,x_2;t_1-t_2) =\int\phi[x]\ \mathcal{D}x $$
where the integral is over all paths that satisfy $x(t_i)=x_i$, with $i=1,2$.

The wavefunction, finally, is given by:

$$\Psi(x,t)=\int_{-\infty}^{+\infty} K(x,x',t-t')\Psi(x',t')\ \mathrm{d}x' $$ which is essentially a convolution.

The path integral formulation of Quantum Mechanics makes the emergence of classical mechanics particularly transparent. In a heuristic sense, one notices that paths differing by a small amount $\delta x$ will lead to actions differing by a small amount $\delta S$. If $\delta S\gg\hbar$, then these paths will add destructively, which means that in the classical limit $\hbar\to 0$, only those paths that make the action stationary, $\delta S=0$, will be relevant to the path integral. Thus, the classical path, which satisfies the classical equations of motion, is naturally singled out.

Prerequisites to learn Quantum Mechanics:

Phys: Newtonian Mechanics; Classical Mechanics; Hamiltonian formalism.

Math: Linear algebra; Fourier Transformation; Partial Differential Equations (PDE); Operator Theory and Hilbert spaces; Lie algebras. Maybe also: Finite Groups, Discrete Groups, Lie Groups, and their representation theory; Young Tableaux.

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