Lecture3 (and part of lecture 4).

Transcription

1 Lecture3 (and part of lecture 4). Angular momentum and spin. Stern-Gerlach experiment Spin Hamiltonian Evolution of spin with time Evolution of spin in precessing magnetic field. In classical mechanics M = [r p]. The angular momentum M Can take arbitrary value. For charged particle rotating around, the angular momentum is proportional to magnetic moment. The current is I = e v, where ( e) is the electron charge, v is 2πr the velocity and r is the radius of the orbit. The magnetic moment of the current μ = I πr 2 = evr 2. Both the angular momentum J = mvr and the magnetic moment are directed perpendicular to the plane of the orbit. We therefore have μ = e J m (SI units). In Quantum mechanics, angular momentum is quantied to integers, in much the same way as energy of a hydrogen atom is quantied to discrete levels. More than that, in addition to integer orbital angular momentum, particles also can be characteried by integer or halfinteger spin S, which is sometimes pictured as intrinsic angular momentum. Both J and S are vector quantities. For integer M (Total angular momentum, JS), the projection onto a given axis is also integer or half integer, and can take 2M1 integer (half integer) values: M, M-1,.-M1, -M. From this fact, follows an interesting property for the square of total angular momentum vector M: M 2 = M(M 1). Indeed, these 2M1 integers (half-integers) characterie projection on all three axes, x,y,. The total square is therefore, e.g., M 2 = 3 < J 2 >= 3 M2 (M1) 2 (M) 2 =M(M1). 2M1 Therefore, projection M is always less than square root of the square of angular momentum. That means that spin or angular momentum is never fully aligned with any, e.g, axis. The smaller the magnitude of angular momentum or spin the stronger the effect. For example, for angular momentum 1000, M 2 ~ However, for spin ½, s 2 = 3/2. The mathematical way to describe this property is introducing the (matrix) operators of angular momentum. For spin ½ these are the Pauli matrices σ = [ ], σ x= [ ], σ y= [ 0 i i 0 ],and the spin operators are defined as s i = ħσ i.it is impossible to determine all

2 three projection of spin (or angular momentum) simultaneously for a given system (particle), in contrast to classical angular momentum. Mathematically this corresponds to different projection operators not commuting with each other, e.g.: s x s y s y s x = is, In general, s i s j s j s i = is k ε ijk, where ε is the Ricci (Levi) symbol, distinct from ero only if all three indices are diffirent, positive when the orderof indices is x,y, or formed by its cyclic permutations, and negative when the order of indices is yx or formed by its cyclic permutations. Stern-Gerlach experiments. Particles with spin pass through the device ( blue rectangle) in which (schematically showm on the left) there is magnetic field and a gradient of magnetic field in, say, -direction. Particles are collected by a red glass screen, and the spatial distribution of points where atoms collide with the screen are measured. The potential energy of particles associated with their magnetic moment is U = μ B. Magnetic field gradient results in a force, U. In classical physics, particles can be characteried by the magnetic moment of arbitrary magnitude. Therefore one would expect continuous distribution of points on the red screen, corresponding to deviations of atoms of the atomic beam from the straight passage through the device caused by gradients for different magnetic moments. However, for focused beam of silver atoms prepared in an oven and directed through the area with magnetic field and its gradient, one observes two sharp spots where silver atoms collide with the screen. That shows that the beam of silver atoms splits into two beams. Thus, silver atoms are characteried by two discrete values of magnetic moment. One can filter the beams by inserting a screen blocking the passage of one of the splitted beams. One gets then a filtering of magnetic moment. For, example, on the left figure, only the lower beam reaches the red screen.

3 Modified Stern-Gerlach device. In a simple SG device, filtered beams propagate at an angle with an original beam of atoms (ions). It is possible to modify the Stern-Gerlach experiment in such a way that the outgoing beam will be parallel to the incoming beam. That would correspond to splitting and then recombining the original beam. For example, one can imagine three magnets, one magnet with a gradient of field in positive -direction, then a twice longer magnet with a gradient of magnetic field in negative -direction, and then again a magnet identical to the first magnet. It is desirable to construct a device in which beams will be splitted into two and then recombined in such a way that two beams experiencing splitting inside the device propagate in identical conditions, so that they are not affected in any way that distinguishes one beam from another. Series of Stern-Gerlach filters. Using Stern-Gerlach filters, one can demonstrate the properties of spins (angular momenta) and their projections. We will denote SG filter in the following way: in the case when the upper of two beams splitted inside the device is closed as in the figure on the previous page; in the case when the lower of the splited beams is closed (and the upper propagates). We will denote the SG device in which beams are splitted and recombined and none of them is stopped. For complete description of the Stern-Gerlach device, we will use notation, where denotes the direction of magnetic field and its gradient. In general we have filters, and, where S denotes the coordinate axis (x,y, or any axis forming some angle S S S with these). Important Stern-Gerlach experiments. Filters in series. transmission is ero. x transmission is non-ero x x transmission is nonero transmission is nonero

4 x x transmission is 0 (*) transmission is 100% (**) Relation of Stern-Gerlach experiments to qubits SG Filter is related to the projection on the eigenstate of σ with eigenvector 0 1. SG Filter is related to the projection on the eigenstate of σ with eigenvector 1 0. SG Filter x is related to projection on the eigenstate of σ x, 2 2 (1 1 ). SG Filter x is related to projection on the eigenstate of σ x, 2 2 ( 1 1 ). Qubit is a linear combination of any two orthonormal vectors. SG filter can measure a state of a qubit flying through the filter. Passing through three SG devices, with the middle device transmitting everything, signifies important quantum mechanical procedure of expansion into base states. Physically, the middle device in the last two experiments above does nothing. One can see that if two side SG devices filter different states, the total transmission is ero. If they filter the same state, the transmission of the beam is 100%. The middle device serves, however, a nice tool: We can write that for the experiment (*), we start from a state, > and can go through two states x,> or x,-> of the open middle device. After these beams passing the modified SG middle device are recombined at its exit they finally pass through device,->. The notation x,> means a state of the divice, and mathematically is a column vector, or in Dirac terms, ket. Analogously, <-, is a string vector, in Dirak terms called bra The probability of passing from, > to x,> is written as a product <,,x>. In terms of vectors of quantum states, this means scalar product of the two vectors. Similarly, probability of passing from x,> to the third device,-> is the scalar product <x, -,>. The total amplitude of propagation in (*) example is a sum of two terms, <, -,x><x,- -,><,,x><x, ->. This sum of two amplitudes corresponds to interference of two paths of transmission. In(*) the interference is destructive, in (**) the

5 interference is constructive. Note that interference proceeds over all available states of the middle device. Here, these are all base states. The middle device in (*) or (**) does nothing indeed. However, we can think about processes in which the vectors of state of ions are manipulated. Imagine for example an SG experiment - --->, in which the magnetic field direction and the direction of its a(t) gradient is changed with time t. This manipulation can be conveniently described as a(t) Note that the second and the fourth SG devices do nothing (no filters) except allowing a convenient way to write down the amplitude of this process: ij <, i, ><, i A j, ><, j, >, where A is the unitary matrix describing the operation of rotating magnetic field orientation. <, i A j, > i is nothing but the matrix element of operator A in the basis of eigenstates of σ. Operator A is one of the operators allowing manipulations on qu-bits. The task is to be able to set the qubit into any possible linear combination of base states and to manipulate qubits by transforming them into arbitrary linear combination of base states. Postulates of Quantum mechanics. Associated to any isolated physical system is a state space. The system is described by a state vector. Properties of base states <i j>=δ ij. Properties of amplitudes < γ β >= < γ i >< i β > i (Insertion of non-filtering SG device with magnetic field directed along the axis that defines base states i) The quantum-mechanical evolution of the state vector is described by an unitary transformation The time evolution is described by a Schroedinger equation iħ Ψ t = HΨ, where H is the Hamiltonian operator. Spin Hamiltonian in magnetic field. In classical physics, the energy of electron with magnetic moment μ is μh. In quantum mechanics, the Hamiltonian describing the behavior of spin

6 Is H = μσh. For time-independent magnetic field, this Hamiltonian gives two eigenstates with energies ±μh, where H is the magnitude of magnetic field vector. The eigenvectors depend on the orientation of magnetic field. For example, when Magnetic field is H = (H x, 0, H ), the eigenvectors are 1 ( 2H(HH ) H x H H ) and 1 2H(H H ) ( H x H H ). Time evolution of spinors (eigenvectors). iħ Ψ t = HΨ. When the Hamiltonian is time-independent, the approach is Ψ = exp { iet ħ }ψ. That gives a time-independent equation for ψ. Now, consider time-dependent magnetic field and time-dependent Hamiltonian.H = μσh, where H = (H cosωt, H sinωt, H ). The spinor is time-dependent, Ψ(t) = ( a(t) ).The system of equations ( )for a(t)and b(t) has the form. b(t) iħa = μh a μh exp(iωt) b iħb = μh b μh exp(iωt) a Let us introduce a = exp(iωt/2)a, b = exp(iωt/2)b. Then we have a system of equations (we drop tilde below) Here γ 1 = μh ħ ω 2, γ 2 = μh ħ. iħa = γ 1 a γ 2 b iħb = γ 1 b γ 2 a Assuming intial conditions spin up, i.e., a(0) = a (0) = 1, b(0) = b (0) = 0, we obtain the following solution of (*) : Ψ(t) = 1 {[(Ω γ iωt 1) exp(iωt) (Ω γ 1 ) exp(iωt)]exp ( ) 2 2Ω 2iγ 2 sinωtxp ( iωt ) }, 2 where Ω = γ 1 2 γ 2 2. We see that there is a probability to flip spin. The amplitude is b(t), and the probability is

7 W = ( γ 2 Ω )2 sin 2 Ωt. This probability is maximal when H = ωħ, i.e. in resonance, and may 2μ become unity at Ωt = π. 2 Applying pulses of certain duration, one sets different values of the qubit.