Challenges¶
Challenge 1: Explore how to define desired states¶
Create the following droplets using the menu
:![]() |
![]() |
![]() |
![]() |
?
?
?
?
Challenge 2: Explore how to define desired states¶
Create the following droplets using the menu
:![]() |
![]() |
![]() |
?
?
?
Solution 2¶
Magnetization Vectors to get an opaque droplet).
(turn off the preferenceFig. 218 I−1¶
Fig. 219 I+1I+2¶
Fig. 220 2⋅I−1I−2I−3¶
Challenge 3: Explore how to define desired states¶
Create the following droplets using the menu
:![]() |
![]() |
![]() |
?
?
?
Challenge 4: Explore how to interpret droplets¶
Using the menu Cartesian product operators.
, create the operator 2Qx(I1,I2) and determine how it can be expressed as a linear combination ofSolution 4¶
The double-quantum x operator involving spins I1 and I2 corresponds to the following droplet:

Fig. 224 ρ0=I1xI2x−I1yI2y¶
Select the menu Fig. 224.
to see the operator decomposition. The Cartesian terms will be displayed in the shorthand notation for Cartesian operators as shown inHence, 2Qx(I1,I2)=0.5(2I1xI2x)−0.5(2I1yI2y)=I1xI2x−I1yI2y.
Challenge 5: Explore how to interpret droplets¶
Using the menu 2I+1I+2I+3 and determine how it can be expressed as a linear combination of Cartesian product operators.
, create the operatorSolution 5¶
The +3-quantum operator 2⋅I+1I+2I+3 corresponds to the following droplet:

Fig. 225 ρ0=2⋅I+1I+2I+3¶
Select the menu Fig. 225.
. The following terms will be displayed in the shorthand notation for Cartesian operators as shown inHence,
Challenge 6: Pulse sequence design challenge¶
Suppose in the course of a pulse sequence, you have created the state I1zI2z−I1yI2y. How can you transfer this state into ±2 quantum coherence by a single pulse?
Solution 6¶
Using the menu I1zI2z−I1yI2y. The corresponding DROPS representation should look like:
, create the operator
Fig. 226 I1zI2z−I1yI2y¶
The droplet representing the bilinear terms involving spins I1 and I2 already has the desired shape and color of ±2 quantum coherence, except that it is not correctly oriented! Rotating it by 90° around the y (or −y) axis will yield the correct orientation, i.e. a non-selective 90°y (or a 90°−y) pulse yields the desired ±2 quantum coherence. This can be tested by selecting
and the resulting DROPS representation at the end of the pulse is shown on the right. (Note the rectangle at the bottom, which represents the pulse, where the blue color represents the pulse phase −y.)
Fig. 227 ρ0=I1zI2z−I1yI2y¶
Challenge 7: Define a new operator¶
Display the droplet corresponding to the operator 2I1xI2x+i2I1zI2x. (When the droplet is oriented properly, you will see the SpinDrops logo.)
Solution 7¶
Using the menu 2I1xI2x+i2I1zI2x. Orienting the
resulting droplet in the Drops display such that the view direction is
along the y axis yields the SpinDrops logo :-)

Challenge 8: Rotations of operators and droplets¶
Starting from the term 2I1xI2y, find a sequence of non-selective rotations (90±x, 90±y, or 90±z) to incrementally create all of the bilinear operators of the form 2I1aI2b with a≠b and a,b∈x,y,z.
Solution 8¶
![]() |
90°_x \rightarrow |
![]() |
90°_z \rightarrow |
![]() |
90°_y \uparrow |
90°_y \downarrow |
|||
![]() |
\leftarrow 90°_x |
![]() |
\leftarrow 90°_x |
![]() |
Challenge 9: Recognizing droplets of antiphase operators (A)¶
Determine for each of the following droplets whether it represents antiphase coherence with respect to the first spin (±2I_{1z}I_{2a} with a ∈ \{x,y\} ) or with respect to the second spin (±2I_{1a}I_{2z} with a ∈ \{x,y\} )!
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Solution 9¶
This challenge can be solved by simply observing the “kissing beans’ head tilt” (c.f. Head-tilt Rule).
right tilt ![]() Fig. 228 ±2I_{1a}I_{2z}¶ |
left tilt ![]() Fig. 229 ±2I_{1z}I_{2a}¶ |
left tilt ![]() Fig. 230 ±2I_{1z}I_{2a}¶ |
right tilt ![]() Fig. 231 ±2I_{1a}I_{2z}¶ |
left tilt ![]() Fig. 232 ±2I_{1z}I_{2a}¶ |
right tilt ![]() Fig. 233 ±2I_{1a}I_{2z}¶ |
Challenge 10: Recognizing droplets of antiphase operators (B)¶
Determine for each of the droplets the exact form of the corresponding antiphase operator!
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Challenge 11: Symmetry of antiphase operators under 180° rotations (A)¶
Verify that the following symmetry relations of anti-phase operators under 180° rotations are faithfully represented by the corresponding antiphase droplets!
2I_{1x}I_{2z} \xrightarrow{180°x} -2I_{1x}I_{2z} |
2I_{1x}I_{2z} \xrightarrow{180°y} 2I_{1x}I_{2z} |
2I_{1x}I_{2z} \xrightarrow{180°z} -2I_{1x}I_{2z} |
2I_{1y}I_{2z} \xrightarrow{180°x} 2I_{1y}I_{2z} |
2I_{1y}I_{2z} \xrightarrow{180°y} -2I_{1y}I_{2z} |
2I_{1y}I_{2z} \xrightarrow{180°z} -2I_{1y}I_{2z} |
Solution 11¶
![]() Fig. 240 2I_{1x}I_{2z}¶ |
180°_x \rightarrow |
![]() Fig. 241 -2I_{1x}I_{2z}¶ |
![]() Fig. 242 2I_{1x}I_{2z}¶ |
180°_y \rightarrow |
![]() Fig. 243 2I_{1x}I_{2z}¶ |
![]() Fig. 244 2I_{1x}I_{2z}¶ |
180°_z \rightarrow |
![]() Fig. 245 -2I_{1x}I_{2z}¶ |
![]() Fig. 246 2I_{1y}I_{2z}¶ |
180°_x \rightarrow |
![]() Fig. 247 2I_{1y}I_{2z}¶ |
![]() Fig. 248 2I_{1y}I_{2z}¶ |
180°_y \rightarrow |
![]() Fig. 249 -2I_{1y}I_{2z}¶ |
![]() Fig. 250 2I_{1y}I_{2z}¶ |
180°_z \rightarrow |
![]() Fig. 251 -2I_{1y}I_{2z}¶ |
Challenge 12: Comparing the speed of coherence transfer¶
Compare the time required to transfer the initial state I_{1x} to I_{2x} in a homonuclear two-spin system when using either a sequence of delays and pulses (such as in a fully refocused INEPT) or an isotropic mixing sequence (TOCSY) for a given coupling constant J12. Which sequence is faster and by how much?
Solution 12¶
For J12 = 1 Hz (and v1 = v2 = 0 Hz), simulations are shown for the sequence 1/(2J12) - 90°x - 1/(2J12) (Fig. 252) and for isotropic mixing (Fig. 253). The isotropic mixing sequence only requires half the amount of time compared to the INEPT-type transfer, i.e. it is twice as fast. The times shown in these examples are not related exactly by a factor of two (1.025 s and 0.5 s) because the 1.025 s includes the 0.025 second 90°x pulse duration, from a pulse amplitude of 10 Hz, but in a real experiment would be extremely short due to much higher actual pulse amplitudes (typically on the order of 1 kHz - 20 kHz).

Fig. 252 On-resonance Magnetization Transfer Sequence¶

Fig. 253 Isotropic Mixing Sequence¶