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For a normalized dilation ${\psi }_{\lambda }(x)={\lambda }^{d/2}\psi (\lambda x)$ and a homogeneous potential of degree $-n$, the energy is ${\lambda }^{2}T+{\lambda }^{n}V$. Stationarity at an eigenstate gives $2T+nV=0$.

If $V_1\ge V_2$ pointwise for Hamiltonians with the same kinetic term, using the ground state of $H_1$ as a trial state for $H_2$ gives $E_1\ge E_2$.

For the trial state $e^{-\alpha x^2/2}$, the dimensionless Rayleigh quotient isIts unique minimizer isand at the minimum $\epsilon ({\alpha }_{\ast })=2+2{\alpha }_{\ast }/3-1/{\alpha }_{\ast }$.

A confining sextic potential energy can have an exactly known ground state. Introduce a length scale $\ell >0$ and energy scale $ϵ={\hslash }^2/(2m{\ell }^2)$. Forthe normalizable wavefunction isThe version with $\ell =1$ uses the corresponding length unit; powers of a dimensional position cannot otherwise be added without scale factors.

To verify the result, set $y=x/\ell$. Then ${\partial }_y^2{\psi }_0=(y^6-3y^2){\psi }_0$, so the Schrödinger equation gives $H{\psi }_0=2ϵ{\psi }_0$. It is the lowest eigenstate by the nodeless theorem for a one-dimensional ground state.

There is also a direct positivity proof. DefineIntegration by parts on the bound-state domain gives $H=Q^{†}Q+E_0$, with $Q{\psi }_0=0$. Since $⟨Q^{†}Q⟩=\Vert Q\psi {\Vert }^2\ge 0$, no normalized state can have energy below $E_0$.

For a regular one-dimensional confining potential, the bound-state eigenfunctions can be ordered by their number of nodes. The unique nodeless normalizable eigenfunction is the ground state.

The spectral arc length $\theta (U)$ is the length of the shortest closed unit-circle arc containing every eigenvalue of $U$.

If $Uv=\lambda v$ and $U$ is unitary, norm preservation gives $|\lambda |=1$.

The numerical range collects the possible expectation values of a linear operator on unit vectors. For a general operator it lies in the complex plane; for a Hermitian operator the values are real.

The numerical range of an operator $A$ is

For $A=\mathrm{diag}(0,1)$ and $|\psi ⟩=a|0⟩+b|1⟩$, the value is $|b{|}^2$. Every point in $[0,1]$ is therefore attained, although the only eigenvalues are $0$ and $1$. This example separates an expectation value from a possible sharp quantum measurement outcome. It also explains why numerical range is useful in unitary gate discrimination: a zero overlap can arise from cancellation among different eigenvalue phases.

Perfect discrimination is possible exactly when some unit vector satisfies

The units of a ring $R$ form a group under the ring's multiplication, denoted $R^{\times }$. Closure follows because a product of units has inverseThe identity element is the ring's one, associativity is inherited, and each multiplicative inverse is again a unit. The reversed order in the displayed formula is essential in a noncommutative ring.

For example, ${\mathbb{Z}}^{\times }=\{1,-1\}$ and $(\mathbb{Z}/8\mathbb{Z}{)}^{\times }=\{1,3,5,7\}$. In the second group, each nonidentity element has square one. For a field $k$, the unit group of the matrix ring $M_n(k)$ is the general linear group ${\mathrm{GL}}_n(k)$, connecting the abstract definition with invertible matrices.

Every height-one prime ideal in a unique factorization domain is principal and generated by an irreducible element. Consequently, two elements contained in the same height-one prime have a common irreducible factor.

Given a family $A$ of approximating functions, a best uniform approximation to $f$ is an element $p\in A$ satisfying

Contracting two-mode oscillator bilinears with the Pauli matrices,gives $[T^a,T^b]=i{\epsilon }_{abc}{T}^c$. The subspace with total occupation $N$ is the spin-$N/2$ irreducible representation and has dimension $N+1$.

For bosonic modes satisfying $[a_i,a_j^{†}]={\delta }_{ij}$, the number-preserving bilinears $T_{ij}=a_i^{†}{a}_j$ obey

For the curvature conventionthe independent nonzero mixed components areand the Ricci scalar is $R=-2/a^2$.

Every radial coordinate line $\varphi =\text{constant}$ is a geodesic. No nonconstant coordinate circle $r=r_0>0$ is a geodesic.

As $x\to 0$, the limit $\sin x/x\to 1$ gives $\sin x\sim x$, while the Taylor series gives $\arcsin x\sim x$ and $\cos x\sim 1-x^2/2$.

These identities supply the formulas used in the tangent half-angle substitution. The half-angle identities include

The tangent is the trigonometric function $\tan x=\sin x/\cos x$ where $\cos x\ne 0$.

For a real angle $\theta$, $\cos \theta$ is the horizontal coordinate on the unit circle reached by a counterclockwise rotation through $\theta$ radians. For an acute angle in a right triangle, it is the adjacent side divided by the hypotenuse. Basic values includeAt $\theta =2\pi /3$, the horizontal coordinate is $-1/2$ even though the vertical sine coordinate remains positive. A positive angle need not produce a positive cosine.

Reflection across the horizontal axis leaves this coordinate unchanged, giving $\cos (-\theta )=\cos \theta$: cosine is an even function. A full turn gives $\cos (\theta +2\pi )=\cos \theta$. For real arguments, its range is $[-1,1]$ and its zeros of a function occur at $\theta =\pi /2+k\pi$. A quarter-turn shifts between the two coordinate traces, with $\cos \theta =\sin (\theta +\pi /2)$.