Intro & PIB Flashcards
Position Operator in x direction
X̂
2 non-zero wavefunctions orthogonal when
integral of the product of them between infinity and -infinity = 0
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modulus of the wavefunction squared is
positive and real
An observable is
a measureable property (e.g. bond length, dipole moment, KE)
A Hamiltonian is
total energy operator (sum of kinetic energy operator T and potential energy operator V)
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What is h bar?
h/2pi
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de Broglie equation
λ = h/p where p=mv
constraints of wavefunctions
finite
single valued
continuous
differentiable twice
wavefunction x conjugate must be integrable over all space
Born interpretation
probabilistic interpretation of the wavefunction
Born interpretation assumes
the wavefunction is normalised (probability of finding the particle somewhere along the x direction is 1)
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momentum operator in x direction
B̂ = p̂(x) = ℏ/i d/dx
kinetic energy operator
T(x) = 1/2 mv² = (mv)²/2m
sub in p̂(x)=mv
sub in p̂(x) = ℏ/i d/dx
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general eigenvalue equation
B̂f = bf
(where the operator B̂ acts on the eigenfunction f to regenerate f multiplied by the eigenvalue b (a constant))
Schrodinger equation for a free particle moving in the x direction
ĤΨ(x) = TΨ(x) + VΨ(x) = EΨ(x)
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Schrodinger equation for free particle moving in x direction with zero potential energy
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Wavefunction solutions to the Schrodinger equation for free particle moving in the x direction with zero potential energy and the energies associated with them
Ψₖ(x) = Aeⁱᵏˣ +Be⁻ⁱᵏˣ
Eₖ = k²ℏ² / 2m (operate on Ψₖ(x) with KE operator)
Schrodinger equation for a particle between the walls moving in a one-dimensional box
same as for free particle as V(x) = 0
wavefunction 0 outside the box
at walls the particle has infinite potential energy - impossible so the particle cannot exist outside the box so wavefunction must be 0.
Continuity of the wavefunction requires it to vanish just inside the box at x=0 and x=L (boundary conditions).
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What are the allowed wavefunctions for a particle in a box?
Ψₖ(x) = Aeⁱᵏˣ + Be⁻ⁱᵏˣ
Ψₖ(x) = A(coskx + isinkx) + B(coskx - isinkx)
Ψₖ(x) = (A + B)coskx + (A - B)isinkx
C = A + B, D = (A - B)i
Ψₖ(x) = Ccoskx + Dsinkx
Ψₖ(0) = Ccos0 + Dsin0
know Ψₖ(0) = 0 so C = 0
Ψₖ(x) = Dsinkx
Ψₖ(L) = DsinkL
know Ψₖ(L) = 0 but D≠0 or Ψₖ(x) would be 0 everywhere
sinkL = 0
kL = nπ, n = 1,2,3…
therefore
Ψₙ(x) = Dsin(nπx/L), n=1,2,3…
How can the normalisation constant D be worked out?
integral of Ψ²ₙ(x) = 1
wavefunctions and energies of the particle in a box
Ψₙ(x) = √(2/L) sin(nπx/L) for 0≤x≤L
Eₙ = n²h² / 8mL, n=1,2,3…
What is the normalisation constant D?
√(2/L)
zero point energy
irremovable energy which means the particle is never stationary
nodes that Ψₙ(x) has
n-1
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