By Pavel Cejnar
This e-book represents a concise precis of nonrelativistic quantum mechanics for physics scholars on the collage point. The textual content covers crucial issues, from normal mathematical formalism to express functions. The formula of quantum conception is defined and supported with illustrations of the overall innovations of simple quantum platforms. as well as conventional issues of nonrelativistic quantum mechanics—including single-particle dynamics, symmetries, semiclassical and perturbative approximations, density-matrix formalism, scattering idea, and the idea of angular momentum—the booklet additionally covers glossy matters, between them quantum entanglement, decoherence, size, nonlocality, and quantum info. historic context and chronology of easy achievements can also be defined in explanatory notes. excellent as a complement to school room lectures, the booklet may also function a compact and understandable refresher of hassle-free quantum idea for extra complex scholars.
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Additional resources for A Condensed Course of Quantum Mechanics
Operator exponential defined through the Taylor series ˆ = e−iA = Uˆ −1 i Example: Uˆ = ( 01 10 ) Eigenvalues u1 = 1 = ei0 and u2 = −1 = eiπ Eigenvectors | + 1 ≡ √12 ( 11 ) and | − 1 ≡ √12 +1 −1 π +1 −1 Aˆ = 0| + 1 +1| + π| − 1 −1| = with +1 −1 2 ∞ (iπ)k k! 1 [X, 2! 3! ˆ Cˆ [X, 0 ˆ Cˆ [X, 1 ˆ Cˆ [X, 2 ˆ Cˆ [X, 3 1 k! ˆ Cˆ +. . 1 [X, 2! 3! ∞ Yˆ =e + k,l=1 1 k! l! ∞ k=0 1 k! ˆ ˆ eY [X, k ˆ Yˆ l [X, k ˆ ˆ [X, ˆ Yˆ ]] = [Yˆ , [X, ˆ Yˆ ]] = · · · = 0 Special case: [X, ˆ ˆ ˆ 1 ˆ ˆ ⇒ eX+Y = eX eY e 2 [X,Y ] Unitary transformations as “quantum canonical transformations” Unitary operators materialize transitions between alternative QM representations, defined by distinct bases in the system’s Hilbert space (see Sec.
N +1 = 2nr + l + 1 radial quantum number = number of nodes of Rnl (r) 1,2,3... 37 Example (c): attractive Coulomb field (hydrogen atom) 2 2 . 6 eV V (r) = − α r Bound states energies & wavefunctions (for the derivation see elsewhere): 2 En = − M2 α2 1 n2 n = 1, 2, 3, · · · ≡ principal quantum number : n = nr + l + 1 nr = 0, 1, 2 · · · ≡ radial q. number = num. of nodes of Rnl (r) Level n degeneracy n−1 l = 0, 1, . . (n − 1) ⇒ dn = (2l + 1) = n2 m = −l, · · · + l l=0 Rnl (r) ∝ ρl e−ρ/2 L2l+1 n−l−1 (ρ) with ρ ≡ Lji (ρ) ≡ 2 na r, where a = dj ρ di i −ρ dρj e dρi (ρ e ) 2 αM ≡ Bohr radius and ≡ associated Laguerre polynomials Graphical expression of oscillator and hydrogen selection rules for quantum numbers Historical remark 1926: Erwin Schr¨odinger presents a series of 4 papers introducing wavefunction and explaining the quantization of energy in terms of the eigenvalue problem.
28 Operators of spin components along x, y, z axes in H ≡ C2 Sˆx = 2 ( 01 10 ) Sˆy = 2 0 ) Sˆz = 2 ( 10 −1 0 −i +i 0 σ ˆx Pauli matrices σ ˆz σ ˆy Projection to general direction n = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) |n|2 =1 nz nx ny ˆ ˆ = Sˆn = n · S = 2 (n · σ) 2 nz nx −iny nx +iny −nz = cos ϑ e−iϕ sin ϑ e+iϕ sin ϑ − cos ϑ 2 Eigenvalues of spin projection Sˆn Det 2 nz −λ nx −iny nx +iny −(nz +λ) =0 ⇒ λ2 = 1 ⇒ sn = Eigenvectors of spin projection Sˆn Eigenequation nz nx −iny nx +iny −nz α± β± α± β± =± Orthogonality: ( α−∗ ∗ β− ) sn = + 2 = α+ β+ −2 has ∞ solutions: nz = ±1 (otherwise solutions known) ⇒ α± = − Normalized solutions: +2 e−iϕ cos ϑ2 sin ϑ2 nx −iny nz ∓1 β± sn = − 2 = −e−iϕ sin ϑ2 cos ϑ2 =0 Projectors to eigenspaces: α ± ∗ ∗ Pˆ±n = β± ( α± β± ) = e+iϕ 2 e−iϕ 2 ϑ 2 cos2 sin ϑ sin2 +iϕ −e 2 Unnormalized eigenvector: |sn = + 2 = − ϑ 2 sin ϑ sin ϑ sin2 ϑ2 −iϕ − e 2 sin ϑ cos2 ϑ2 nx −iny nz −1 for sn = + 2 for sn = − 2 | ↑ + | ↓ with z = e−iϕ cot ϑ 2 z ⇒ z ≡ stereographic projection of vector n2 onto C Any superposition |ψ =α| ↑ +β| ↓ represents a state of spin pointing in a fixed direction n, which is obtained from z = α/β by the stereographic projection.