Square Barrier Barrier with EV 0 n What
Square Barrier ØBarrier with E>V 0 n What is the classical motion of the particle? 1
Square Barrier Ø In regions I and III we need to solve Ø In region II we need to solve 2
Square Barrier Ø The solution in Region I contains the incident and reflected wave Ø The solution in Region III contains the transmitted wave Ø The solution in Region II is 3
Square Barrier Ø As usual we require continuity of ψ and dψ/dx at the boundaries n n At x=0 this gives A and B in terms of C and D At x=L this gives C and D in terms of E Ø The results are 4
Square Barrier Ø We define reflection R and transmission T coefficients Ø And I’ll leave it to you to show that R+T=1 5
Square Barrier Ø Using relations for k and k. II, we can rewrite the transmission coefficient T as 6
Square Barrier Ø There is one interesting feature n n n With E and V 0 fixed, the transmission coefficient T oscillates between 1 and a minimum value as the barrier width is varied We call the wave in the case of T=1 a resonance A resonance is obtained when k. IIL=nπ w This means T=1 at values of L=λ/2 in region II w That is, a standing wave will exist in region II 7
Square Barrier ØBarrier with E<V 0 n What is the classical motion of the particle? 8
Square Barrier Ø In regions I and III we need to solve Ø In region II we need to solve 9
Square Barrier Ø The solution in Region I contains the incident and reflected wave Ø The solution in Region III contains the transmitted wave Ø The solution in Region II is 10
Ø We could again apply. Barrier boundary conditions on Square ψ and dψ/dx Ø But it’s easier to note the difference between this case and the one previous is Ø Thus for E < V 0, T becomes 11
Square Barrier 12
Square Barrier ØThus we get a finite transmission probability T even though E < V 0 n n This is called tunneling You can think of tunneling in terms of the uncertainty principle w As shown in Thornton and Rex, when the particle is in region II, the uncertainty in kinetic energy is V 0 – E w The uncertainty in energy is comparable to the barrier height and there is a probability that particles could have sufficient energy to cross the barrier 13
Square Barrier Ø For κL >> 1, the tunneling probability T becomes Ø For rough estimates we can further approximate this as (see example 6. 15 in Thornton and Rex) Ø The exponential shows the importance of the barrier width L over the barrier height V 0 14
Scanning Tunneling Microscope 15
STM Ø Invented by Gerd Binnig and Heinrich Rohrer in 1982 Ø Nobel prize in 1986! Ø The basic idea makes use tunneling n n When a sharp needle tip is placed less than 1 nm from a conducting material surface and a voltage applied between them, electrons can tunnel between the tip and surface Since the tunnel current varies exponentially with the tip-surface distance, sub-nm changes in distance can be detected 16
STM 17
STM Ø STM tip 18
STM Ø Tunneling through the potential barrier 19
STM Ø Raster scanning with constant Z Ø Raster scanning with constant tunneling current 20
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STM 22
STM Ø The STM tip is attached to piezoelectric elements (usually a tube) that precisely control the position in x-y-z n n Used to control tip-surface distance (z) Used to raster scan (x-y) 23
STM Ø STM can also be used to manipulate atoms via van der Waals, tunneling, or electric field forces TIP 24
STM 25
STM 26
Quantum Corrals Ø Electron in a corral of iron atoms on copper 27
Quantum Corrals Ø Electron in a corral of iron atoms 28
Alpha Decay Ø Geiger-Nuttall law n Nuclei with A > 150 are unstable with respect to alpha decay w An alpha particle (α) consists of a bound state of 2 protons and 2 neutrons (4 He nucleus) w A(Z, N) → A(Z-2, N-2) + α w Effectively all of the energy released goes into the kinetic energy of the α 29
Alpha Decay Ø Geiger-Nuttall law n Radioactive half lives vary from ~10 -6 s to ~1017 s but the alpha decay energies only vary from 4 to 9 Me. V 30
Alpha Decay Ø Geiger-Nuttall law n n The experimental data follow the Geiger-Nuttall law A calculation of the quantum mechanic tunneling probability explained this law and was one of the early successes of quantum mechanics 31
Alpha Decay 32
Alpha Decay Ø Calculation of the decay probability W Ø Do this for 238 U alpha decay with r. N=7 F and Tα =4. 2 Me. V 33
Alpha Decay Ø Calculation of P and ν 34
Alpha Decay Ø Calculation of transmission probability T n Preliminaries w Calculate the height of the Coulomb barrier w Calculate the tunneling distance 35
Alpha Decay Ø The Coulomb barrier is not a square well n n There is a way in quantum mechanics to calculate T correctly (called the WKB approximation) but for today we’ll just estimate the equivalent height and width of a square well Use VC=20 Me. V and r’=25 F 36
Alpha Decay Ø Calculation of T 37
Alpha Decay Ø Calculation of t 1/2 38
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