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1

de Broglie relation

λ = h/p = 2π/k

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2

Einstein relation

E = hν = ħω

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3

correspondence principle

at high enough energies quantum mechanics agree with classical mechanics

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4

basis of QM

i) quantum state of particle characterized by wavefunction containing all the information possible to obtain

ii) Ψ is interpreted as the probability amplitude of a particle’s pressence (|Ψ|² = probability density)

iii) principle of decomposition applies to measurements

iv) the equation of the wavefunction must be linear and homogeneous

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5

principle of decomposition

-result must belong to set of eigen results

-each eigenvalue a is associated to an eigenfunction Ψₐ whose measurement always results in the eigenvalue

-the probability of finding the eigenvalue a at t=t₀ is |cₐ|/Σ|cᵢ| (comes from superposition Ψ = ΣcᵢΨᵢ)

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6

Schrödinger equation

Ĥ Ψ(**r**,t) = E Ψ(**r**,t)

Ĥ = - ħ²/2m ∇² + V

E = iħ ∂/∂t

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7

TISE

obtained through separation of variables

- ħ²/2m ∇²Φ(**r**) + V Φ(**r**) = E Φ(**r**)

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8

boundary conditions for wavefunction

continuous

single-valued

normalised ∫|Ψ|²=1

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9

bound solutions

specify all 3 BC

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10

unbound solutions

ignore the normalisation BC

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11

square well of width a with infinite walls

Ψ = sin(nπx/a) for even n

Ψ = cos(nπx/a) for odd n

εₙ = n²π²/a²

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12

dirac-delta potential V = - α δ(x)

Ψ = √(mα)/ħ e^(-mα/ħ² |x|)

ε = -mα²/2ħ²

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13

harmonic potential V = mω₀²x²/2, H = p²/2m + mω₀²x²/2

εₙ = (n + 1/2) h ν₀

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14

rigid rotor

εₗ = ħ² l (l+1) / 2I

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15

hydrogen atom

εₙ = -13.6 eV / n²

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16

scalar product of wavefunctions

(Φ, Ψ) = ∫ Φ* Ψ d³r = (Φ, Ψ)*

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commutator

[A, B] = AB - BA

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18

orthonormal basis

basis if every element in the space can be expressed as linear combinations of the elements, verified by closure ∫di **u**ᵢ(**r**) **u**ᵢ*(**r’**) = δ(**r**-**r**’)

orthonormal if (**u**ᵢ (**r**), **u**ᵢ,(**r’**)) = δ(i-i’)

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19

projection operator

Pₐ = |a> <a|

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20

closure relation

∫ da |uₐ><uₐ| = 1

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21

trace

tr(A) = Σᵢ <i|A|i>

same regardless of basis

tr(AB) = tr(BA)

same regardless of basis

tr(AB) = tr(BA)

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22

properties of hermitian operators

eigenvalues are real

eigenvectors corresponding to different eigenvalues are orthogonal

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23

hermitian operators

A = A†

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24

observable

a hermitian operator whose eigenvectors form a basis of the state space

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25

3 theorems of commuting observables [A,B]=0

if |Ψ> is an eigenvector of A then B|Ψ> is an eigenvector of A with the same eigenvalue

if |Ψ₁> and |Ψ₂> are eigenvectors of A, <Ψ₂|B|Ψ₁>=0

iff they commute, they share eigenvectors

if |Ψ₁> and |Ψ₂> are eigenvectors of A, <Ψ₂|B|Ψ₁>=0

iff they commute, they share eigenvectors

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26

<r|p>

e^(i **p**·**r** / ħ) / (2πħ)^(3/2)

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27

momentum operator

**p̂** = - i ħ **∇**

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28

position operator

**r̂** = **r**

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29

angular momentum operator

**^L** = **r̂** × **p̂** = -iħ (**r** × **∇**)

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postulates of QM

(i) the state of a physical system at a fixed time is defined by a specifying a ket belonging to the state space

(ii) every measurable physical quantity can be described by an observable acting on the state space

(iii) the only possible result of measuring an observable is one of its eigenvalues

(iv) when the physical quantity A is measured on a system in the normalised state |Ψ>, the probability of obtaining the eigenvalue aₙ is Σᵢᵍ |<uₙ₍ᵢ₎|Ψ>|² where g is the degeneracy of the state and u the eigenvectors of aₙ

(v) if the quantity A is measured to be aₙ, the state of the system immediately after measuring is the normalised projection of the wavefunction onto the subspace associated with aₙ Pₙ|Ψ>/√(<Ψ|Pₙ|Ψ>)

(vi) the time evolution of the state vector |Ψ(t)> is governed by the S.E. iħ d/dt |Ψ(t)>/ = Ĥ |Ψ(t)>

(ii) every measurable physical quantity can be described by an observable acting on the state space

(iii) the only possible result of measuring an observable is one of its eigenvalues

(iv) when the physical quantity A is measured on a system in the normalised state |Ψ>, the probability of obtaining the eigenvalue aₙ is Σᵢᵍ |<uₙ₍ᵢ₎|Ψ>|² where g is the degeneracy of the state and u the eigenvectors of aₙ

(v) if the quantity A is measured to be aₙ, the state of the system immediately after measuring is the normalised projection of the wavefunction onto the subspace associated with aₙ Pₙ|Ψ>/√(<Ψ|Pₙ|Ψ>)

(vi) the time evolution of the state vector |Ψ(t)> is governed by the S.E. iħ d/dt |Ψ(t)>/ = Ĥ |Ψ(t)>

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31

expectation values

<A> = <Ψ|A|Ψ>

constant of motion: conserved if it commutes with the hamiltonian and does not depend on time

iħ d/dt <A> = <\[A,H\]> + iħ <dA/dt>

constant of motion: conserved if it commutes with the hamiltonian and does not depend on time

iħ d/dt <A> = <\[A,H\]> + iħ <dA/dt>

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32

ΔA

√(<A²> - <A>²)

\

\

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33

Ã

A - ΔA

ΔÃ = ΔA = √<Ã²>

ΔÃ = ΔA = √<Ã²>

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34

operator â

√(mω/2ħ) x̂ + i/√(2ħmω) p̂

annihilation operator: â |n> = √n |n-1>

unless n<1 in which case â |n> = |n>

annihilation operator: â |n> = √n |n-1>

unless n<1 in which case â |n> = |n>

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35

operator â†

√(mω/2ħ) x̂ - i/√(2ħmω) p̂

creation operator: â† |n> = √(n+1) |n+1>

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operator N

a† a

has eigenvalues ≥ 0

if |n> is an eigenvector of N:

-if n=0, a|n> = 0

-if n>0, a|n> = (n-1)|n>

-â† |n> = (n+1)|n>

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37

Lₓ

obtained from **L** = **R** × **P**

Lₓ = RᵧP₂ - R₂Pᵧ

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general definition for angular momentum

any set of observables such that

[Jₓ , Jᵧ] = iħ J₂

[Jᵧ , J₂] = iħ Jₓ

[J₂ , Jₓ] = iħ Jᵧ

therfore J² = Jₓ² + Jᵧ² + J₂² commutes with all Jᵢ

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39

raising operator for linear momentum J₊

J₊ = Jₓ + i Jᵧ

J₊ |j, m> = √(j(j+1) - m(m+1))ħ |j, m+1>

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40

lowering operator for linear momentum J₋

J₋ = Jₓ - i Jᵧ

J₋ |j, m> = √(j(j+1) - m(m-1))ħ |j, m+1>h

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41

hamiltonian hydrogen

P²/2m + L²/2mr² + V

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