Applications of Quantum Physics Module:2 Applications of Quantum Physics Particle in a 1-D box (Eigen Value and Eigen Function), 3-D Analysis (Qualitative), Tunneling Effect (Qualitative) (AB 205), Scanning Tunneling Microscope (STM). Arpan Kumar Nayak (Ph.D) School of Advanced Sciences (SAS) Particle in a 1-Dimensional Box Time Independent Schrödinger Equation 2 d 2 V ( x) E 2 2m dx Region I Region II Region III KE PE TE Wave function is dependent on time and position function: V(x)=∞ K.E=0 0 V(x)=0 V(x)=∞ K.E=0 L 0 V(x)=0 for L>x>0 V(x)=∞ for x≥L, x≤0 Classical Physics: The particle can exist anywhere in the box and follow a path in accordance to Newton’s Laws. Quantum Physics: The particle is expressed by a wave function and there are certain areas more likely to contain the particle within the box. 1 ( x, t ) f (t ) ( x) 0 Time Independent Schrödinger Equation x 2 d 2 ( x) V ( x) E 2 2m dx Region I and III: Applying boundary conditions: 2 d 2 ( x) * E 2 2m dx Region II: 2 d 2 ( x) E 2m dx2 0 2 Finding the Wave Function 2 d 2 ( x) E 2m dx2 d 2 ( x) 2m 2 E dx2 This is similar to the general differential equation: d 2 ( x) k 2 2 dx A sin kx B cos kx So we can start applying boundary conditions: x=0 ψ=0 0 A sin 0k B cos0k x=L ψ=0 0 Asin kL A0 where n= Calculating Energy Levels: k 2 2 E 2m 2mE k 2 2 E n h 2 L 2m4 2 2 2 2 h 2 E 2 k 2h2 E 2m4 2 2 nh 8mL2 II A sin nx L But what is ‘A’? Normalizing wave function: L ( A sin nx / L) 2 dx 1 0 L x sin 2(nx / L) A 1 2 4 ( n / L ) 0 n sin 2 L 2L L A 1 n 2 4 L 2 0 0 B *1 B 0 kL n Our new wave function: * Since n= 2 L A 1 2 * A 2 L Our normalized wave function is: II 2 nx sin L L Case-1 n=1 E1 2 h 2 8mL E0 1 min, x 0, L 1 max, x L / 2 1 2 x sin L L Case-2 n=2 E2 2 4h 2 8mL 4 E1 2 2 min, x 0, L / 2, L 2 max, x L / 4,3L / 4 2 2x sin L L Particle in a 1-Dimensional Box II 2 nx sin L L Applying the Born Interpretation II 2 2 nx sin L L n=4 n=3 E x/L 2 n=4 E n=3 n=2 n=2 n=1 n=1 x/L Particle in a 3-Dimensional Box 2 nxx X ( x) sin Lx Lx 2 nyy Y ( y) sin Ly Ly 2 nzz Z ( z) sin Lz Lz 8 nxx nyy nzz (r ) sin( ) sin( ) sin( ) V Lx Ly Lz 2 2 2 2 h nx ny nz Enx , ny , nz ( 2 2 2) 8m Lx Ly Lz Operator • A rule that transforms a given function into another function Identifying the operators Linear Operator • A linear operator has the following properties A f 1 f 2 A f 1 A f 2 Acf c A f Eigen Function and Eigen value In any measurement of the observable associated with the operator , the only values that will ever be observed are the eigenvalues ‘a’ which satisfy the eigenvalue equation: This is the postulate that the values of dynamical variables are quantized in quantum mechanics. A f ( x) kf ( x) f(x) is eigenfunct ion of A with eigen valu e k Eigen function and eigen value f x e ikx Is it eigen function of momentum operator ? What is eigen value ? Eigenvalue equation Eigenvalue equation (Operator)(function) = (constant factor)*(same function) ^ Example: eikx is an eigenfunction of a operator Px = -ih F(x) = eikx = -i h eikx x = -i2 hk2eikx = h k2eikx Thus eikx is an eigenfunction x Tunneling Effect Tunneling Effect The quantum particle exhibits wave-like nature, it can reflect and transmit (tunnel) through the potential barrier Barriers and Tunneling (Tunnel Effect) Consider a particle of energy E approaching a potential barrier of height V0, and the potential everywhere else is zero. First consider the case of the energy greater than the potential barrier. In regions I and III the wave numbers are: In the barrier region we have Reflection and Transmission The wave function will consist of an incident wave, a reflected wave, and a transmitted wave. The potentials and the Schrödinger wave equation for the three regions are as follows: The corresponding solutions are: As the wave moves from left to right, we can simplify the wave functions to: Probability of Reflection and Transmission The probability of the particles being reflected R or transmitted T is: Because the particles must be either reflected or transmitted we have: R + T = 1. By applying the boundary conditions x → ±∞, x = 0, and x = L, we arrive at the transmission probability: Tunneling Now we consider the situation where classically the particle doesn’t have enough energy to surmount the potential barrier, E < V0. The quantum mechanical result is one of the most remarkable features of modern physics. There is a finite probability that the particle can penetrate the barrier and even emerge on the other side! The wave function in region II becomes: The transmission probability that describes the phenomenon of tunneling is: Tunneling wave function Consider when κL >> 1 then the transmission probability becomes: This violation of classical physics is allowed by the uncertainty principle. The particle can violate classical physics by DE for a short time, Dt ~ ħ / DE. Scanning Tunneling Microscopy The scanning tunneling microscope was developed at IBM Zürich in 1981 by Gerd Binning and Heinrich Rohrer who shared the Nobel Prize for physics in 1986 because of the microscope. Iron atoms on the surface of Cu(111) Scanning Tunneling Microscopy Developed on the principle of quantum mechanical tunneling Components 1 metal tip 2 Piezoelectric scanner 3 current amplifier 4 Bipotentiostat 5 feedback loop 6 detector Tip and sample must be electrically conductive (metals) Advanced STM Ultra high vacuum and low temperature STM set-up Graphene Graphene STM working Principle STM works on the principle of quantum mechanical tunneling Tunneling current I A eV e 2 2 m 2 d A is constant e is electron charge V is voltage m is mass of electron Φ is work function of metal tip d is distance between tip and sample STM working procedure Constant current mode Image the surface with constant tunnel current and variable height Feed back loop help to maintain constant current Surface (height) structure can detect Constant height mode Image the surface with constant height and variable tunnel current Electron density on the surface can detect Surface of different structures Resolution : 0.1 nm It is resolve individual atoms in a materials STM images of gold and Si crystal STM Applications Widely used in nanotechnology Image the surface structure Estimate surface roughness 3D images of the surface Locate the defect on the surface of crystal Understand electric structure of materials